Daniel Plainview seems to have only one real drive; the greed and ambition to grow. As he says, he sees the worst in people, and wants to be away from them. He goes on elaborate lengths to be outstanding, including playing a father for a long time so as to make his business seem more likable. He deceives people, gives good speeches, and cares about image. He values hard work, and he's seen transforming from a man with a broken leg with no one to help in the middle of a desert to an oil tycoon.
Eli was just a would-be reverend in a small superstitious village. Considering how both men were deceitful and dishonest in their own faith—Eli was only religious because it brought him the respect of the normal village folk, and Daniel was a father mostly because of the business—and they saw through the lies of each other, they weren't initially different and they both knew it. Eli moves up a bit too, and he likes the attention, but he's broke.
Why does Daniel pick his most bitter fight with Eli?
It can't be because he saw himself in Eli, because he didn't. Eli became a pitiful broke drinker, while Daniel ended up wealthy, albeit pitiful, and Daniel started to hate him from very early on. He didn't choose him for the blessing for the crowd to see, and he lashed out at him when H.W. was hurt in an accident. Daniel is shown not to take a liking to the superstitious show Eli is putting on early on, so it might be his aversion to religion, though I doubt it, as religion didn't seem to be one of the main focuses in the movie. Lastly, there are others much easier to hate in a movie about the dark side of a man taking over who is in the oil industry. He hates the lot of them of course, but they don't become his arch nemesis; they don't become people he longs to take revenge on years later.