How do general practitioners screen for suicide risk?

Listen. Ask. During your physical, we ask about general signs of depression. If yes, then we check out how bad, including "ever think of hurting yourself?" during regular visits, we listen for clues about stress and how it effects you. Always more to your whole story.
Patient interview. This is done by reviewing a patient's social history as well as past mental health history. Family history is also quite important for evidence of mental health disorders and suicidal behaviors in family members. However, outright questioning a patient if they have thoughts of suicide or self injury or a history of hurting themselves is the best way to screen for suicide risk.