I believe Wickham had to leave the regiment over debts (especially the gambling debts, I imagine). He needed to go somewhere where he could hide out, and London was a good place for that -- he came to the regiment originally from London, so he might know people there who can help him hide.

He may have taken Lydia along for a couple of reasons: (1) she was a companion who would tell him how wonderful he was (a boost to his bruised ego); (2) she was an energetic young female who didn't seem to have high moral standards, and he could have some fun while he was running & hiding; (3) she had some spending money, and she could provide the funds for the trip.