Sorry, but legally a non-starter. In fact, he could run for VP, and she ceased to be able to be Pres, he would elevate to that position.
No person shall be ELECTED to the office of the President more than twice, and no person who has held the office of President, or acted as President, for more than two years of a term to which some other person was elected President shall be elected to the office of the President more than once. But this article shall not apply to any person holding the office of President when this article was proposed by the Congress, and shall not prevent any person who may be holding the office of President, or acting as President, during the term within which this article becomes operative from holding the office of President or acting as President during the remainder of such term.
Bill Clinton is ineligible to the office of VP.
See Amendment 12:
...But no person constitutionally ineligible to the office of President shall be eligible to that of Vice-President of the United States.