[The President] shall have Power, by and WITH the advice and consent of the Senate, to make treaties, provided two thirds of the Senators present concur; and HE shall nominate and by and WITH the advice and consent of the Senate, SHALL APPOINT Ambassadors, other public Ministers and Consuls, JUDGES OF THE SUPREME COURT...
Remember, all of this is written about the powers of a "Legal" President, not a usurper.
The Constitution states that whoever receives a majority of the votes of the electors “SHALL BE THE PRESIDENT.” (12th Amendment).
270 electoral votes are a majority and Obama got 365 electoral votes in 2008 and 332 electoral votes in 2012. A Joint Session of Congress meets in January to count and certify those electoral votes and it only takes the written objection of one Representative and one Senator to stop the certification of electoral votes until both Houses of Congress resolve the objection by majority vote with one vote per state in each House.
Obama’s electoral votes were certified without written objection, twice.
If there had ever been a court ruling of ineligibility or a congressional resolution of ineligibility, you would have constitutional ground to stand on. But there has been no such finding. 21 court rulings have declared Obama to be eligible for the position he holds and Congress has sent him hundreds of bills to sign into law and Congress has confirmed hundreds of his nominees.
For example, 113 federal judges nominated by Obama have been confirmed unanimously. The last Obama federal judges to be confirmed are Jill Parrish to the District Court for Utah and Jose Rolando Olvera, Jr. for the Southern District of Texas. They were both confirmed 100-0 on May 21, 2015.