Well given that Oswald was born in 1939, with a POD of 1940 that shouldn't be too difficult.
Let's say that after being put into a foster home during 1942-1943 due to his mother's poverty, Oswald, instead of returning to his mother is put up for adoption due to his mother being unable to care for him (perhaps she dies?). Lee Harvey Oswald is later adopted in 1944/45 by a caring stable family who go one to have a profound effect on his later life. Despite being dyslexic, Lee is able to pass high-school (though not without some difficulty.)
After graduating high-school Lee decides to follow after the example of his adoptive father and join the armed forces, namely the USMC and proceeds to rise through the ranks. Eventually Oswald is sent to Vietnam where he serves with distinction, winning a Medal of Honor for his gallantry in combat. Upon returning to his native Louisiana, Oswald opens up a small construction business before being lured into politics.
Running as a State Republican, Oswald soon rises to the top of the Louisianan Republican party winning the nomination in 1980, the election, and going on to serve two terms. Come 1988 he's a strong contender for the GOP presidential nomination and he's eventually picked to balance the ticket with Robert Dole.
Dole wins the election over an incumbent Democrat (?) and proceeds to serve two terms, overseeing the fall of the Soviet Union. 1996 sees Oswald receive the GOP nomination and win a narrow victory.
One of Oswald's most memorable moments as President would be the attempted assassination attempt by JFK Jr. who, in a delirious state, shot Oswald claiming he had "ruined [his] political fortunes".
(Obviously, US history on a whole is going to be a whole lot different. Assuming that the lone gunman theory is correct, in TTL JFK's visit to Dallas is rather uneventful. Personally, I doubt JFK would have pulled out of 'Nam but that's just my opinion. I think if anything he'd follow a similar course to Johnson, attempting to push through a Great Society while continuing to ramp up Vietnam. After JFK, I doubt RFK would run in 1968, maybe a bit later. Perhaps Johnson or Humphrey runs in 1968 losing perhaps to an up and coming Ronald Reagan?)