Just before Elvis is drafted someone figures out Colonel Tom Parker is in the country illegally and he gets deported. In the ensuing chaos Elvis is unable to make any of the movies or record any of the songs he would have before going into basic training. Over the next two years his star fades while (thanks to an avoided plane crash) Buddy Holly, Ritchie Valens, and maybe an unknown rise, and come 1960 when he would have mustered out he sees his music career as over and opts to stay in the Army.*
Politics and world events trundle along on their more or less OTL course for awhile longer, and ultimately by the mid-60's the United States finds itself bogged down in Vietnam. Elvis is given a field command and deployed to South Vietnam, (a place he'll call home until 1973) steadily rises through the ranks until he makes Colonel, and makes quite a name for himself back home. Back home in the States a two term JFK (I'm throwing him a bone and letting him live) oversees a period of unrest as the civil rights and a somewhat weaker counter culture movement flare up, and in 1968 is succeeded by Richard Nixon. Nixon's first term goes about like it did in OTL, and thanks to not being dumb enough to do Watergate in TTL he has a marginally successful second term where he ends US involvement in the Vietnam war, eeks through a healthcare deal (something he was working on), and few other things.
Getting back to Colonel Presley, with the end of the war Elvis retires from the service and returns home to Tennessee where the local Democratic Party (he voted for more Democrats than Republicans), wanting to take advantage of a returning popular war hero, persuades him to run for governor. Elvis is of course reluctant, but is talked into it ("You always liked helping people, so think of all the good you could do in Nashville.") and goes on to wins the 1974 gubernatorial election, and reelection in 1978.
On the national stage Nixon leaves office in 1977 with an economy that is really beginning to splutter, and after a close election he is succeeded by fellow Republican Ronald Reagan who is beset with a vary Jimmy Carter-esque term. In 1979 Elvis, not a fan of Reagan's policies or the direction the country has gone under him, throws his hat into the ring of Democratic primary, and thanks to his legacy as a successful two term governor he is able to clinch his party's nomination. Election night 1980 proves to be a rather long one with several states remaining to close to call until after midnight, but when the votes are counted the result is.....
.....And thus the King of Rock and Roll is the President.
*I'm thinking he gets clean and stays clean while in the Army.