First of all, Paul's appeal is far far too narrow to make it past a Republican primary through any conventional means.
Realistically speaking, the only route for Paul to make it past the convention involves a delegate-heavy focus where he wins the caucus states and succeeds in stuffing delegate slots even in primary states with his supporters, though they may be bound to other candidates, forcing a deadlocked convention with a few other candidates, and then being anointed after delegates are unbound.
This is the strategy Paul and his supporters employed in 2012 to some small success. But the key thing is that if Paul wins, he won't be doing it with the support of a majority or even a small plurality of Republican voters. He'll have achieved it through mastery of rules lawyering, stacking the deck in contests thought inconsequential with little turnout, etc. Establishment Republicans, social conservatives, almost everyone except his supporters would feel utterly cheated and robbed (remember how Obama's supporters warned in 2008 against having the super-delegates anoint Hillary? Think that times 100), would feel like the party had been hijacked, and there would be a lot of evidence to back it up. I'd expect a third party candidacy in that case, an utterly split Republican party, and consequentially a Democratic presidential landslide.
The other avenue which I'm not completely sure I believe involves having a very different 2000 such that Democrats enter many wars and Paul's anti-war platform gains more traction among Republicans. This is the avenue SLCer partly uses in the (excellent) TL Bridge to the 21st Century (Paul doesn't actually get the nomination there, but he comes very close.) It's still very hard for me to see Republicans nominating an anti-war candidate in that case, however.