Orderic Vitalis wrote that Malcolm was betrothed to Margaret of Wessex, the daughter of Edward the Exile. (That's Margaret in the illustration, reading the Bible to Malcolm. She was later declared a saint.) When they married in 1070, this made Malcolm the brother-in-law of Edgar Ætheling, one of the potential heirs to the throne of England in 1066 after Edward the Confessor's death. This was the context for Malcolm supporting Edgar with troops in Edgar's attempts to capture the throne from the more successful claimant, William II of Normandy.
You can see here and here how Edgar relied on Malcolm for support and simply as a place of refuge when things got bad. Malcolm's good relationships were with English royals, not the Normans who invaded in 1066. His aid to Edgar helped prompt William to engage in the Harrying of the North, a series of military campaigns by the new king of England to devastate much of the north of England to punish for past and deter future attempts to coalesce around Edgar Ætheling and use him to replace William.
Malcolm and William finally met in 1072 and negotiated a treaty. Malcolm pledged his loyalty to William, which meant little: Malcolm continued raiding in Northumbria until Williams sent his son Robert Curthose in 1080 at the head of an army. This time Malcolm kept the peace to which he had agreed.
In 1091, when Robert Curthose started rebelling against his brother William Rufus (who had inherited after William's death), Malcolm did not take sides. He did, however, take advantage of their fighting to go south and besiege Newcastle and the castle built by Curthose. This was a step to pushing Scotland's boundary further south. An approaching English army led by Rufus motivated Malcolm to retreat north.
His final dealing with royalty came when he marched south with his son to discuss the return of lands given to Malcolm by William II. Rufus refused, saying that the matter would be settled by the English barons. Malcolm rejected this idea and returned to Scotland, gathering an army and ravaging Northumbria more harshly than ever before. On 13 November 1093, he was ambushed by the Earl of Northumbria, Robert de Mowbray, and killed.
His interment includes a story that needs to start with his wife, Margaret. Tomorrow I'll tell you about the queen who became a saint.