You have 2 questions here, and so there are 2 answers to provide.
Your first question, Why did Walter want to kill Alucard, has little to no real revealed reasons. The Wiki article for Walter C Dornez Has one quote that offers Alucards belief why.
Alucard goes on to present what he believes is the reason for Walter's betrayal; his fear of becoming old and useless. To prove his abilities to himself, Walter desired to destroy Alucard, and allowed this obsession to consume him.
This however has one flaw, in that Walter planned to die regardless of success or not in the end. proving oneself to not be obsolete, but then becoming obsolete instantly afterwards is counterproductive. Why he planned to die is unknown, but it was guaranteed for reasons I will mention later. This means that this reasoning was not the truth when Walter attacked, but it may have been the reasoning when he began. It was heavily hinted that Walter began his betrayal when he was about 14 years old, and first witnessed Alucard's power. It could be that he had that kind of reasoning at that time, but later on changed his mind, wanting to surpass or kill Alucard regardless of his death after success. That would satisfy half of the origional goal of beating Alucard, and in his older age allow him to die, at least satisfying his obsession.
It could also be that he saw Alucard as a real monster, a threat, and decided Alucard had to die, one way or another.
A third reason is hinted at by one of his quotes
"We are the evening's entertainment. And I... just wanted to do something with my time on the stage worthy of applause..."
if this one is the truth, it could be that he did have an obession with killing Alucard, but it was so that he could have the Legacy of being the one who killed Alucard the Neigh immortal and undeniably evil monster, to feel that his life truly had meaning.
We never truly got a straight answer out of him though, So this will probably forever be shrouded in that mystery unless someone else has stuff I missed.
As for your second question, How he got younger, it is a fairly simple one. He had the Millennium Doctor perform a special version of the Vampirification Surgery on him. He became an Artificial Vampire , and what made it special was that it's modified but flawed regeneration regresses his body as it activates. That tidbit is hidden in the wiki in the third paragraph of Walters Millennium Secret weapon part of his history Section. The down side was that it was not a perfected surgery, and was rushed, so permanent damage was done to him. He was literally deteriorating as he fought Alucard. His deterioration would cause his Flawed regeneration to kick in, and he would heal, and regress in age. As such, the surgery doomed him to only have hours or days left alive, no matter what he did.