Hmm. If he's always been a Republican, then if we want his background to be the same, we'd need the Chicago political machine to be Republican. (Note that the infobox still places him from Illinois.) That'd change so much that I think it'd butterfly away at least half our modern politicians.
Another idea I had is that he gets his start in local Chicago politics, as a reformer - a Republican who cleans up the machine. That would give him a lot more substantiation for his message of "change"; I'm pretty sure he could make the jump directly from mayor of Chicago to President.