Basically, the President has to nominate a new ambassador and the Senate has to confirm the nomination. Article II Section 2 of the US Constitution states:
[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, and all other Officers of the United States...
The following steps assume that the nominee has already become a Foreign Service Officer (for career diplomats) or is close to the President (for political appointees) and is nominated by the President.
Firstly, the nominee has to fill up all the necessary paperwork which includes "financial disclosure forms, questionnaires from senators, background documents, security forms and more".
"If the decision is made by the State Department, in coordination with the White House, to nominate you for an ambassadorship, the next thing that happens is you get about a half a truckload of paper," he noted.
Source: State of play: Becoming an ambassador takes time -- and paperwork
Secondly, government officials then vet the documents and the candidate. If nothing goes wrong, they inform the host country of the nominated ambassador (a process called agrément).
Thirdly, if the host country does not object (it is rare that they object), the nomination is submitted to the Senate Foreign Relations Committee and the nominee gets a hearing. Since it depends on the committee whether to hold hearings, nominations sometimes expire at the end of a Congress before they are even considered. Or in other times, hearings are held but delayed for a long period of time.
Fourthly, when the committee approves the nominee through a committee vote, the nomination goes to the full Senate floor for a vote. Again, Senators can force up to 30 hours of floor debate per nominee which can delay the floor vote.
For further information, check out State of play: Becoming an ambassador takes time -- and paperwork.
Here's a simplified timeline of the steps to become an ambassador, taken from the US Embassy to the Holy See site.