Compile using latexmk in emacs

First, look at TeX-command-list to ensure that it has an entry for latexmk. If not, you need to add a line to this variable to allow the executable to be invoked by Auctex; unfortunately this variable is not well-documented (see the manual); I explain a little later.

Second, the default program is specified by the variable TeX-command-default. Change this to the identifier used by TeX-command-list, that is, the string just before the specification of the latexmk command.

E.g., if one of the elements in TeX-command-list is shown (which you can see using C-h v):

 ("Latex Make" "latexmk %(-pdf) %t" TeX-run-TeX)

then TeX-command-default needs to be "Latex Make".

The Auctex variable TeX-command-default

This variable isn't, as I said, well-documented. It should be a list of lists, each of which takes the form:

 '("Handle" "external-command %(switches) %(arguments)" Auctex-handler)

where Handle introduces a handle for facility to be run from the C-c C-c dialog, so the list of handles in TeX-command-default is the list of possible completions at this prompt. The second option is a form specifying the executable to be run, and Auctex-handler is an Elisp function that handles the process and invokes the external command. TeX-run-TeX and TeX-run-command are the most common handlers, but there are special handlers for Bibtex, the spell checkers, etc.