You have two main options here:
The Quick Way: Rent a sod cutter. It removes the grass and the top inch of root/soil, giving you a clean slate immediately.
The Manual Way: Use an edging tool to cut your existing lawn into squares. Lift the squares and flip them, leaving the bare soil up. Tamp the squares down. Repeat until the whole lawn is soil side up.
The Patient Way: "Sheet Mulching." Lay down cardboard over your grass, soak it with water, and cover it with 3-4 inches of soil/compost. Wait a few months for the grass underneath to die back.