I'm not sure there's any reality in which this would even happen. Trump was a Clinton supporter in 2008, so if she won and there was no birther movement, I'm not sure he still would have even been a Republican. Much less a politically active one inclined to run for president as he was in our timeline. Wouldn't hurt that Obama never would have mocked him for not being president at the WHCD.
Also, if Hillary wins in 2008, the political landscape likely looks vastly different. She had a chance at winning states like Arkansas, Kentucky, and West Virginia based on the polls at the time, as insane as that seems today. There was a steep drop-off in WWC Democratic registration following Obama's election, that may not have occurred had Hillary (their preferred candidate at the time) had won. Possibly there is no Tea Party and the backlash to her simply is not as pronounced as it was against Obama. Study after study has showed that backlash was heavily amplified for racial reasons that would not have been as big a factor with a white incumbent, probably not even with Obama as VP. Hillary also might have simply been better at supporting the party in downballot and off-year races, which was one of Obama's weaknesses.
If that's all the case, and Obama still ends up facing Trump in 2016 somehow, I think he easily wins. I don't see any scenario in which Trump beats Obama head-to-head. If the economy is recovering under two-term incumbent President Hillary Clinton as it was under Obama, and Trump is as much of an unpopular jackass as he was in reality, VP Obama easily wins.
I think we'd be still be dealing with some form of the Tea Party because of latent sexism instead of latent racism. It might have been even stronger outside of the south.
