Gatsby is pretty self-conscious isn't he? He is well aware of the relationship between Nick and Daisy being cousins. Gatsby likely believes anything Nick hears, thinks or believes about Gatsby could travel back to Daisy in one way or another.
It is this lack of confidence and outright desire to convince Daisy that he is worthy of her that drives Gatsby to make sure Nick "knows him."
What is interesting about all of the times that Gatsby tries to reveal something personal about himself or as he tries to demonstrate his good ol American boy feel, he ends up lying just a little by not knowing the whole truth. For example, he thinks San Fransisco is in the MidWest... Good try Gatsby. People who lie have good reason to lack confidence.
In Chapter Four of Fitzgerald's The Great Gatsby, Gatsby says to Nick,
Look here, old sport,...What's your opinion of me anyhow?....Well, I'm going to tell you something about my life....I don't want you to get a wrong idea of me from all these stories you hear.
Gatsby asks Nick what he thinks of him so that he can tell his life story in order to prevent Nick's believing the rumors about him. He says to Nick, "This is the God's truth." Gatsby wants Nick to think well of him. This is why he asks Nick "What's your opinion of me anyhow?"
It is after finishing his own personal history, a history that resembles that of the early immigrants, the adventurers of early America who sought "the American Dream," that Gatsby makes his request. He does not ask Nick what he thinks of him in order to make a request because, before doing so, he tells Nick about himself. He does this, not to make the request, but to impress Nick and somewhat endear himself to Nick by recreating an endearing red-blooded American who seeks the Dream. Having endeared himself to Nick will lessen the likelihood of Nick's refusing his request:
'I'm going to make a big request of you today,' he said, pocketing his souvenirs with satisfaction,' so I thought you ought to know something about me. I didn't want you to think I was just some nobody. You see I usually find myself among strangers because I drift here and there trying to forget the sad thing that happened to me.'
The reason that Gatsby does this, in my opinion, is because of what he is going to do later in the chapter. He is going to ask Nick (actually, he is going to have Jordan Baker ask Nick) to set up a meeting between Daisy and himself. He's going to have it happen at Nick's house.
So he is going to have Nick be part of his plan to get Daisy. If Gatsby is going to have Nick do this, he wants to be sure Nick thinks well of him so he will be more willing to do this for him.