Well, I don't think the Republican Party would go for Nixon's Southern Strategy. Thus, perhaps in partnership with MLK, JR or even a moderating Malcom X, the Republicans would remain the party of Civil Rights and classic liberalism. Also, remember that the Republicans had been until rather recently the Party that was generally against foreign intervention. Having endured quiet a long period under FDR, Truman and JFK with only a break of Ike's two terms, the Republicans will probably move back towards their base and support smaller government, capitalism without government interference and indirect instead direct support for non-alliance anti-communists.
As to who would run well, I'm not sure yet. But most likely JFK would still win, especially if he survived an assassination attempt. So in 1968 the Republicans would try to capture as much of the Black and liberal vote as possible. Most likely the their ticket would include a person from the Northeast or Northwest and a person from the West. This way they could win the election without worrying about the Solid South. Of course Wallace would probably still run and drain some votes from the Dems. giving them reason to whine later.
Benjamin
I think the Conservative faction growing within the GOP would turn off folks like MLK and Southern blacks by 64 nonetheless. Likewise, keep in mind that the Republicans had been losing the black vote since FDR and the black vote was rather pro-Democrat, at least in the North, and would likely be pro-Democrat nation wide by 1968 as much as it was OTL following civil rights.
Likewise, the GOP was not the party of small government and noninterference in the "Free Market", etc and that wasn't their base; that was the Conservative faction's base. The party was simply always pro business and focused more on economics. However, the GOP was frequently supportive of Progressivism concerning social and economic matters and up until pretty much the 20's and from the (Franklin) Roosevelt era to Reagan, was rather Progressive dominated or moderate dominated. Likewise, the Conservative faction was frequently criticized for their ideas as not being what the GOP had been about, which gives you a good idea of the party's standing.
In 1968, I think that what you'd see in the GOP is a moderate to Liberal who could likewise draw in the Conservative faction, and I'd think that'd be George Romney. Romney was already prominent and a likely runner, pleased the moderate-liberal faction, and his views on morality and Vietnam (since events would likely ensure he'd be pro-intervention in Vietnam and increased intervention) could please the Conservatives. Likewise, he was popular with labor and blacks and could draw in both (though it'd likely be a flash in the pan for the GOP in those areas since I think that future Republicans wouldn't gain those two; sorta like blacks who voted for Nixon).
I'm also not sure about Wallace. A ticket for Dixiecrats is generally always a possibility in any TL at any point, and perhaps it was destined as a last hurrah for Dixiecrats before they joined the GOP, but I'm not sure if it's definite. Likewise, a Southern Democrat could be on the main ticket in 1968.
But who is the Democratic contender in 1968?