While he was part of the Nation of Islam, Malcolm X had no respect for MLK. Absolutely none. He openly accused MLK of being a paid puppet of whites, promoting integration and the philosophy of non-violence as ways of keeping blacks defenseless. He felt that not only was this a Trojan Horse of sorts, something that would allow white people to continue their injustices against blacks, but also that it letting whites off the hook. Malcolm wanted whites to face true justice, which would be answering for their treatment of blacks. And if carrying out that justice required violence, then so be it.

Like almost every other part of Malcolm's worldview, he softened a lot on this stance after he left the Nation of Islam and traveled to Mecca. After that, he viewed MLK as a respected leader, someone who he could and would work with, but still who was ultimately interested in different goals than his own. MLK wanted peaceful integration, Malcolm still wanted peaceful separation.

MLK would never publicly return the harsh criticism that Malcolm X directed at him. He claimed to have a deep affection for Malcolm, and respected Malcolm's ability to cut straight to the root of the problem, though of course he disagreed with Malcolm's proposed solutions. But he never came anywhere near the level of vitriol for Malcolm that Malcolm had for him.

I apologize for the lack of sources. I'd be happy to provide them another time. Most of this is based on videos/transcripts of Malcolm X's speeches, and the part about MLK's views in return come from a very quick reading of The Autobiography of Martin Luther King.

I wrote a paper for a college English class in 2008 arguing that the public discord between the two (and their movements) was beneficial to the MLK civil rights movement, and found some evidence that at least Malcolm X wanted it that way. I don't think that their public rhetoric is a good indicator of their personal beliefs. Some quotes:

King told interviewers that he felt “Malcolm [X] has done himself and our people a disservice. Fiery demagogic oratory in the Black ghettos, urging Negroes to arm themselves and prepare to engage in violence, as he has done, can reap nothing but grief” (King, 265-6). Malcolm X accused that “Just as Uncle Tom, back during slavery, used to keep the Negroes from resisting the bloodhound or resisting the Ku Klux Klan, today Martin Luther King is just a twentieth-century or modern Uncle Tom or religious Uncle Tom” (Leader, 129). Publicly, there was animosity.

...

Elijah Muhammad prohibited any Black Muslims from participating in politics or protests, as he kept his own work towards achieving a separate state behind the scenes. Malcolm resented this policy, and wished that “in the Little Rocks and the Birminghams and other places, militantly disciplined Muslims should also be there for all the world to see, and respect, and discuss” (Carson, 14).

...

A few months before Malcolm X was shot, he gave a speech in Selma, Alabama, in which he spoke against Martin Luther King. Nonetheless, before he left Selma, Malcolm arranged a meeting with Martin’s wife Coretta Scott King (Martin was in jail at the time). King later told reporter David Halberstam what he had learned of the meeting: “he thought he could help me more by attacking me than praising me. He thought it would make it easier for me in the long run” (Halberstam, 338).

You can read my paper complete with embarrassing typos and professor feedback as well as works cited at https://docs.google.com/document/d/1OxXf_YKNZHxLh7EePa-v6CUYJnNM6Pc7UFxZCekkbKU/edit?usp=sharing

You may also be interested in the FAQ section entitled Malcom X & MLK, to which this has been added.