As I have discussed in the past, heroic characters in the Golden and Silver Age for the most part didn't require a motivation. They fought against crime and injustice simply because they were heroes. This is part of what makes Batman and Spiderman so unique; they did have strong personal motivations. Hawkeye, on the other hand, did have a motivation, but it was an odd one: He wanted the acclaim that comes along with being a hero. Remember, the first we see him is when he's simply a carnival side-show act:
Awhile later, after the apparent death of the Black Widow, Hawkeye decided to try out for the Avengers. He showed his suitability for the team by breaking into their HQ and tying up Jarvis:
There was another aspect of the quarrel between Hawkeye and Cap that was interesting. Despite appearing physically the same age, Cap was from a different generation. Hawkeye was brash and arrogant, while Rogers was cautious and a bit stodgy.
At first, I confess that I found their verbal jousting a bit tedious. Stan Lee's arguing characters tended never to resolve their differences. JJJ always hated Spiderman and the feeling was mutual. The Thing constantly bickered with Johnny and Reed.
But then a funny thing started to happen. Gradually, over a period of several years, we began to perceive that Hawkeye was starting to appreciate Captain America. Here's a hint of it:
And while Hawkeye did not give up the occasional jibes about Captain America being Methuselah, you definitely began to get the feeling that it was pro-forma; that there was no real antipathy between the pair. It was a nice moment of growth and change.