It would be pretty hard to do considering his background and personality. You'd have to have a PoD before 1908, when he suffered a nervous breakdown that stopped him from graduating high school. He was apparently a highly intelligent child, so he could become a lawyer or businessman (the latter is more likely as it would not involve interacting with people in courtrooms and his grandfather, Whipple Van Buren Phillips, was a businessman). From there, it would be relatively easy for him to go into politics. I can see him becoming a Joe Kennedy-style Northern conservative Democrat, probably starting out in state legislature and before moving on to governor/representative/senator. As far as I know, "shaking hands and kissing babies" wasn't as big a part of politics back then, but he was still a severe introvert who suffered from panic attacks, so you'd have to have some major change in his personality for him to be suited for office.
At some point, the Democrats somehow get it into their heads that he would be a good nominee. I guess, as a Rhode Island reactionary, he could appease both Northerners and Southerners in a contested convention. To win, he would have to face a situation like 1932 (i.e. against an extremely unpopular Republican). There is no way that America would ally with the Nazis or a similar regime, but Lovecraft would certainly advocate for neutrality. He would not handle the stress of the presidency well at all, and I can see him dying early in his term.