Well, I doubt he'll be impeached, but if he does run for President does he resign? or does he just delegate governing to his staff? and who in his right mind would want Christie's staff in charge of anything?
Because of campaign finance law, it might be difficult to for him to run for president while remaining in office as governor, without sacrificing a great deal of fundraising $:
https://uselectionatlas.org/FORUM/index.php?topic=176680.0
Citizen's United saved his ass, and SCOTUS could likely weaken campaign finance laws even more.