the Type-Moon Wikia on Arturia compiles alot of the information scattered across the different Type-Moon media. keep in mind that this would be from translations. this answer is more or less a copy and paste of the information from the Wikia
Artoria was born into a time of chaos and war that began with the demise of the Roman empire. It had been believed to be indestructible, but it was only awaiting its destruction at the hands of invading barbarians. In preparation for the war against the barbarians, Rome deprived her island province of any military forces left there. Once Britannia lost the empire's protection, it could not escape becoming independent, causing it to break into smaller countries soon after. This time of barbarian invasions and self-destructive strife between clans started a long period of war that would later become known as the "Dark Ages". Artoria was born into this period as the heir to the throne, the daughter of the King of Britain, Uther Pendragon, and the blue-eyed Igraine, the wife of Uther's servant, the Duke of Cornwall. The king, believing in Merlin's prophecy, yearned for the birth of his appointed successor, but the child born was not the one he desired.
It is said in the legend that Arthur was left in Merlin's care as the price for his assistance in ensuring the success for Uther's love of Igraine, saying "I will properly guide this child, one bearing a great destiny, and protect him from the crisis of the royal family." The true reality was that Artoria was not a boy, so the king could not make a child that was not male his successor even if she was fated to one day become a king. She was entrusted to the king's vassals and was to be raised as the child of a mere knight. The king fell into despair at the situation, but Merlin was delighted because the sex of the one to become king had never mattered. He was confident that the fact of the girl being separated from the castle until the day of prophecy was proof that she would become king.
Merlin placed Artoria in the care of Sir Ector, a simple and wise old knight, who raised her as his adopted child and successor. Though Ector did not believe in the prophecy, he did feel the same air from the girl as he did his king. He felt that he must raise her as a knight, and he wished for her to grow. He never needed to wish for such a thing, as she trained day after day to become stronger than anyone. She swore to bear her sword for the only reason that "only a king can save a ruined country headed for death" without ever being told such. During this time, she was raised with Sir Kay as her brother, but they still loved each other as siblings after she learned the truth of their relationship. She acted as his squire and received training from him, while also doing other chores such as pulling along his horse. She was better than him in terms of swordsmanship, but she never beat him in a fight due to his arguments declaring himself the winner rather than actual skill.
Once the day of prophecy arrived, knights and lords from around the country gathered to be selected as king. Each expected the selection to be through jousting to select the most superior one to become a king, but the only thing prepared at the place of selection was a naked sword stuck in a stone with a golden inscription on the hilt reading "Whosoe'er pulleth out this sword of this stone is rightwise king born of England." While many knights grabbed the sword trying to follow the command, none were able to pull it out. As they began the expected method of selection by jousting, Artoria, only an apprentice not qualified for jousting, neared the deserted stone of selection and reached out for the sword without hesitation.
Before grabbing it, Merlin appeared before her to tell her to think things over before taking it. He told her she would no longer be human upon taking hold of the sword, but she only responded with a nod because she has been prepared for the fact that "becoming a king means no longer being human" ever since she was born. She knew that a king is someone who kills everyone to protect everyone. She thought about it every night and shuddered until morning came. While not one day passed where she did not fear that fact, she said that it would end this day. The sword was pulled out as if it were only natural to do so, and the area was filled with light. She became something not human in that instant. The king's gender doesn't matter, and no one will care about the king's appearance or even notice it as long as the king acts like a king. Even if anybody did notice the king was female, it would not be an issue if she was a good king. This started the time of the king who would become a legend. She stopped aging at 15 years old, back when she pulled Caliburn from the stone.
Artoria led Britain from Camelot, and after becoming a feudal lord like her father, she became a king with many knights under her, including the esteemed Knights of the Round Table. Her growth stopped at that time she pulled the sword because of its magic, so many knights feared it as ominous. Most instead praised their master's immortality as divine. Her battles thereafter were the acts of a god of war. She always led from the front, and no enemies could stand in her way. There was no defeat for a body admired as a dragon in human form. She only knew victory for ten years and twelve battles while she ran through those days as the king. She never turned back and was never disgraced. She was raised as a king and fulfilled her obligations as the king.
Due to the problem of being able to produce an heir, Merlin used magecraft to turn her into a pseudo-male capable of producing sperm for an unknown duration of time. During this period, she was enchanted by her sister, Morgan le Fay, who took some of Artoria's sperm, developed it within her own ovaries, and gave birth to the homunculus and complete clone of Artoria, Mordred. She was born and raised without Artoria's knowledge, quickly growing up due to her accelerated aging, and she managed to join the Round Table through her own efforts and Morgan's recommendation. She worshiped her king while hiding her identity, and she was ecstatic to learn of her heritage.
Upon claiming her heritage and right to the throne, Artoria completely rejected Mordred, and refused to recognize her as an heir. Mordred believed the reason that she was not accepted was due to the hatred her father bore Morgan, and that no matter how much she worked to excel over others, she would forever be viewed as a tainted existence. The love she had for the king up until that point was so great that her rejection made her hatred burn.
She married Guinevere, but it was only out of obligation to have a wife. The truth was that Lancelot, one of Artoria's closest friends, and Guinevere loved each other, and due to Artoria having concealed her gender, Guinevere had to carry the burden of this her entire life. Guinevere felt guilt for her actions, but felt relief that it was Lancelot who had fallen in love with her. He, who shared the same ideals as the king, would share the burden with her and not cause the country to fall into a dangerous situation. He did so, supporting Guinevere and the King from the dark on his unrighteous path.
Artoria needed to act as the son of the king, as the one to govern the many territories and control the knights had to be male. While some people did grow suspicious of her, only her father, Merlin, and Kay knew the exact truth of her identity. She literally covered herself in steel to seal that truth for all her life, and due to her immortality from the protection of the fairies, no one questioned her small body or face that seemed like that of a girl. She was honored by the knights as a good-looking king, thought of as an invincible warrior whose looks or body size had impact on her standing.
During that time, the people living in fear savage invasions only wanted a strong king, and the knights would only follow an excellent commander. Having met these criteria, no one questioned her. She was considered fair and selfless as she always stood at the front of the army while defeating her enemies on the battlefield. While many enemies and civilians died, the king's choices were always considered correct, and she served as the king better than anyone else. No one doubted or had any need to doubt as long as the king was right.
Her army reconstructed the lost cavalry, and knowing no losses, they ran through the battlefield while defeating foreign infantries and crashing through numerous ramparts. Many people were discarded for her to join the battles, and all her enemies had to be defeated once she joined. It was common practice for the military to meet its needs by sucking all the resources out of a local village to supply the battle to protect the country. It can be said that no knight killed more people than her, and it is unknown if she ever found such to be a burden.
She strictly kept to the oath that a king is not human and that one cannot protect the people with human emotions. She never narrowed her eyes in grief while sitting on the throne, and she settled every problem while working hard in government affairs. She managed to balance the country without any deviations, and she punished people without a single mistake. Even after, or possibly because of, winning battles in victory, commanding citizens without disorder, and punishing hundreds of criminals, one of her knights murmured "King Arthur does not understand human feelings."
It is possible everyone felt that way, that the more perfect she became as a king, the more they needed to question her as a ruler. They felt that a human without emotion cannot rule over others, leading to several reputable knights leaving Camelot. She simply accepted this to be a natural event that is part of the process of government, isolating the fair king honored by her knights. Having abandoned her emotions from the start, she did not change her mind even if she was abandoned, feared, or betrayed. There was no right or wrong to someone who saw such events as trivial.
Her final battle for her country began in such a way, and the battle at Badon Hill ended in complete victory. The savages sought reconciliation due to her overwhelming results, and her country that was only awaiting destruction earned a brief period of peace. The country finally began to return to the country of which she had dreamed.
Lancelot's affair with Guinevere was eventually revealed through the plans of traitors who hated Camelot, causing them to stand opposed. Artoria did not see this inescapably unrighteous action as a betrayal, but instead understood Guinevere's sacrifice due to having concealed her gender. She still acted in the capacity of a king, and Guinevere was to be executed. Lancelot could not stand by while she was killed, bringing about a tragedy where no one was right and no one was wrong.
Excalibur's scabbard was stolen while she repelled an assault along her country's borders; when she returned inland, she discovered Britain is being torn asunder by civil unrest. Despite her valiant efforts to placate the dissent, she was mortally wounded by the traitorous knight Mordred during the Battle of Camlann. Her dying body was escorted to a holy isle by Sir Bedivere. Artoria ordered a grieving Bedivere to dispose of Excalibur by throwing it back to the Lady of the Lake; in her absence, she reflected on her personal failures, regretting her life as king. Before her last breath, she appealed to the world; in exchange for services as a Heroic Spirit, she asked to be given an opportunity seek the Holy Grail to save her country.
i emphasize the last bit as after this is when Arturia made her contract with the world and started to be summoned to the Holy Grail Wars. unlike the other Heroic Spirits who are effectively clones of their real selves in the Throne of Heroes, Arturia never made it there which is why she can retain her memories of previous times she has been summoned.
after the events of Fate/Stay Night, depending on the route Arturia' fate different. i summarize it in this answer but to sum that up
In Fate we see Arturia accept things when Sir Bedivere returns for the final time
In -Last Episode- because Arturia terminated her contract with the World by destroying the Holy Grail by her own will she goes to Avalon in the Reverse Side of the World, waiting for Shirou who eventually reunites with her as apart of a miracle Merlin talks about
In Unlimited Blade Works:
In the Good End she remains with Rin as her Servant
In the True Ending Arturia fades away however we don't know if her contract was terminated or not and if it was if she ended up in Avalon aswell
in Heaven's Feel, regardless of the ending Arturia is corrupted by Angra Mainyu and is killed by Shirou close to the final battle. we don't know of her fate after this
in Fate/Hollow Ataraxia she is summoned for Angra Mainyu false Holy Grail War. because Angra Mainyu was trying to recreate the 3rd war Arturia is summoned as if she was summoned by the Edelfelt sisters who used their sorcery trait to summon 2 Sabers (2 sides of the same Hero) Arturia is summoned with the ability to swap with her Alter personality (the Arturia corrupted by Angra Mainyu)
Also while we do not know when about in Arturia's own timeline this occured
- in Fate/Labyrinth she is summoned as Norma Goodfellow's/Manaka Sajyou's Servant during the Subcategory Holy Grail in Caubac Alcatraz's labyrinth. she forms a party with 3 other Servants, Archer (Robin Hood from Fate/Extra), Caster (Medea from Fate/Stay Night) and Assassin (Hassan-i-Sabbah from Fate/Stay Night). because of a fading contract with Norma/Manaka and the uses of her Nobel Phantasm during one battle Arturia fades away but not before saying her goodbyes to the other 3 Servants and entrusting them to give Norma/Manaka a Mystic Code she found