I watched Dragon Ball Z and want to know why Frieza wanted to kill Goku in the first place.
There are a few reasons to answer your question.
As Shaymin Gratitude correctly said, Frieza wanted to kill all the Saiyans. The reason being that the Saiyans are a warrior race. They grow stronger after every fight (Zenkai boost). He knew this. He also knew about the legend of a Super Saiyan. He feared that someday, a Super Saiyan will emerge and defeat him. This led him to destroy planet Vegeta and drive the race to the verge of extinction.
When he was at Namek, he was looking for the dragon balls. He wanted to wish for eternal life. As Goku was also looking for them, they crossed paths, fought and eventually he lost his wish and suffered a defeat at the hands of Goku.
After his humiliating defeat on Namek, he swore that he will have his revenge on Goku.
This is precisely why Frieza is always trying to kill Goku.