As Eugene Gant notes below, Martin Luther King is frequently claimed as either a conservative, or in some cases a member of the Republican Party.
Historically, black support for the Republican party was was common before the 1930s, because Republicans were the Party of Lincoln, just as the white Solid South support for Democrats was because the Democrats were the Party of not-Lincoln.
FDR's presidency changed all that, although FDR did dedicate a monument to Robert E. Lee in Dallas in 1936.
But King, private life aside, wasn't any kind of conservative.
The late Joe Sobran said in 1998 that
"To take just one example, I gnash my teeth when conservatives argue that ‘affirmative action’ violates ‘the spirit of Dr. King’—’color-blind justice,’ and all that. Nonsense. If King were alive today, he’d certain support state-imposed racial preferences. He was a Marxist, always moving leftward. Liberals are right to claim him as their own; conservatives who appeal to his ‘spirit’ only make fools of themselves."[Media and Mythology, By Joe Sobran, SOBRAN’S, November–December 1998]
This is true. In spite of the oft-repeated quote about wanting his children to be judged by the "content of their character" King was in favor of hard racial quotas for the benefit of his people. (When I say oft-repeated, I mean that search of NationalReview.com shows 207 instances of "content of their character" and 91 instances of "content of our character",the title of a Shelby Steele book.)
American Renaissance, on the other hand, knows better.
"King has been praised, even by conservatives, as the great advocate of color-blindness. They focus too narrowly on one sentence in his ‘I Have a Dream’ speech, in which he said he wanted to live in a nation ‘where [my children] will not be judged by the color of their skin but by the content of their character.’ The truth is that King wanted quotas for blacks. ‘[I]f a city has a 30 percent Negro population,’ King reasoned, ‘then it is logical to assume that Negroes should have at least 30 percent of the jobs in any particular company, and jobs in all categories rather than only in menial areas.’"(Emphasis added.) [The Unknown Martin Luther King, Jr., by Benjamin J. Ryan, American Renaissance, January 2009]
The difference between the two ideas of King may come from the fact that Amren.com's Ryan actually read King's last book, Where Do We Go from Here: Chaos or Community? from which the quotation above was taken.
Chapter 5 of that 1967 book has an ambitious program of pressure on local businesses (backed by black economic boycotts and pulpit denunciations) to do their hiring on a hard quota basis, put their money in black-owned banks, and stock products made by blacks.
And of course, this was after the passage of Civil Rights Act of 1965, which made such hiring technically illegal.