Odysseus could not have killed all the suitors without their help, which he requests. You will find the scene of their slaughter in Book XXII. Here is an excerpt:
They therefore aimed straight in front of them and threw their spears. Ulysses killed Demoptolemus, Telemachus Euryades, Eumaeus Elatus, while the stockman killed Pisander. These all bit the dust, and as the others drew back into a corner Ulysses and his men rushed forward and regained their spears by drawing them from the bodies of the dead.