I'll tell you all I know about Khamul, based on the entire Lord of the Rings Trilogy, and the Unfinished Tales.
Khamul was second in command to the Witch-King, and was an easterling while the W-K was a Black Numenorean. That means that either Khamul was an extremely powerful sorceror from the east, or the ruler of a massive kingdom, because it's been stated that the Black Numenoreans wielded great power and status.
As far as information about Khamul in the Trilogy, the Unfinished Tales give him the role of the first Nazgul met in the Fellowship, the one who journeyed west of the Baranduin and questioned the Gaffer, Maggot, and who chased Frodo, Pippin and Sam. He can be quoted with the lines that make up my signature. His importance in the trilogy then, besides being the Witch-King's "number two". . . he was looking for Baggins.
It's widely believed that Khamul was the "Keeper of Dol Goldur", as the Decipher Ulaire Attea card refers to "Nazgul number one" under the Witch-King, which would be him if it had followed the text and not the movies.
Khamul would have at one point been one of the Nazgul tasked with going into the east and 'recruiting' fell eastern armies as servants to the Dark Lord Sauron, along with all the other eastern wraithes.
No doubt, since the wise speculated a Nazgul was still running Dol Goldur at one point, it was likely they thought it was him who was 'polluting' Mirkwood and making it not so nice, before Gandalf went on his stealth run and discovered that Sauron in fact had returned.
When the Witch-King was eliminated, Khamul didn't take over his armies, he went with the other Nazgul on their flights, terrorizing from high above and spreading fear, instead of personally leading ground troops like the somewhat cocky Morgul-lord had done.
Maybe he didn't want some warrior-maiden to spring up and do him in as well.
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"I come from yonder...Have you seen Baggins? Baggins has left, he is coming. He is not far away. I wish to find him. If he passes will you tell me? I will come back with gold." - Khamul the Easterling
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