A psychiatrist is a medical doctor who has gone on to specialize in psychiatric illnesses. They diagnose mental illnesses by using the Diagnostic and Statistical Manual (DSM), which outlines the criteria a person must meet in order to be diagnosed with a certain condition. In order to determine this, they conduct a thorough assessment - they ask a lot of questions about the symptoms a person is experiencing presently, as well as what's happened for that person in the past. By putting all of these clues together, they come up with a diagnosis (or not).
So in diagnosing bipolar disorder, a psychiatrist assesses the person, and gets any additional information they can from family members or close friends. Quite often they have the person complete a questionnaire such as the Mood Disorder Questionnaire (you can find this at www.dbtforbipolar.com), which also helps the psychiatrist to see what specific symptoms of bipolar disorder the person might be experiencing or has experienced in the past.
Bipolar disorder is a diagnosis that has been missed a lot in the past - people don't usually go to the doctor when they're feeling good, which means the doctor often won't find out about hypomanic or manic symptoms unless they ask very specific questions or talk to family. If you're meeting with a psychiatrist for an assessment, just make sure you're honest and you answer the questions to the best of your ability. Whatever the diagnosis is (if there is one), just remember that it doesn't change your experience: you already know how it feels to live inside your body; now you just have a name to call that experience by.