I agree with what was written by others below, but I think there are a couple of points that haven't been touched on that might be confusing you.

First, Wickham was getting to the point where he had to leave Brighton, anyway. (From Mrs. Gardiner's letter: "He confessed himself obliged to leave the regiment on account of some debts of honour which were very pressing....") Wickham didn't leave to marry Lydia; he was going to have to leave, anyway and Lydia made herself available, so he took her along. It has been speculated, as mentioned below, that part of the motive was financial, and that's possible.

Second, I think it is misleading to way that Wickham "wanted to elope," if that suggests that Wickham left Brighton with a plan to marry Lydia, though I don't know right off how else to refer to it, other than as an elopement. Denny, at least, didn't believe that any marriage was really planned. It was only through Darcy's intervention that he did marry her.