My boss told me I was getting a promotion at the start of this last pay period. My paycheck did not reflect this. When I reached out to him he said he "forgot" to promote me. He says I will be compensated for it on my next paycheck. Should I reach out to his boss to ensure that this happens?
Only if you think your boss was deliberately lying to you. It doesn't sound that way to me, so I'd leave it another month, and see what happens.
Wait and see
Your manager messed up. He admitted it, and promised to correct his mistake. If you give him the chance to correct his mistake, it will show that you trust him, and strengthen your relationship.
If, at this stage, you harass him about his mistake, it will strain your relationship. If go over his head and involve his superior, it will strain that relationship even further. The more your relationship is strained, the more your boss has reason to doubt himself about deciding to promote you in the first place.
Only act if the boss doesn't deliver
Even if you don't see that promotion next paycheck, don't involve your boss's boss. He doesn't know that your boss promised you a promotion. Speak only to your own boss, and make it clear that you expect him to do what he has promised.
Write an email, with a request for receipt if possible. This proves you did your due diligence in asking, and provides him with a reminder come next pay period. That said, I would personally wait until next pay period to come and see if he did as asked. Give a chance for him to make good unless you have a legitimate reason to be concerned that he is lying to you.