Ronald has a deep-seated need to be attached to someone. There's a gaping void in his life which only Alex (Chris McCandless) can truly fill. An only child, whose mother and father were also the only children in their respective families, Ronald feels he's pretty much reached the end of the line now that his wife and son are no more. (They were tragically killed by a drunk driver.)
Ronald is the last of his family; he knows that when he dies, his family will die with him. So he asks Chris if he can formally adopt him as his grandson as a way of keeping the family going. But that's not the only reason why Ronald makes such an unusual request. It's clear that the old man thinks of Chris as a kind of replacement son for the one he so tragically lost. More to the point, Chris has come to be the grandson that Ronald has never had.
Ronald is further drawn to Chris by the fact that he thinks—wrongly, as it turns out—that he doesn't have any family, either. In his eyes, that makes them kindred spirits who are similarly bereft, lost souls in need of a family.