How does Mark Antony manage to achieve success by the end of the play?
Antony achieves success but Shakespeare hints that this success may be short lived (as it was in real life--Octavius and Antony only joined forces to defeat Cassius and Brutus; when that was over, Antony fled to Egypt where he stayed with Cleopatra and got his own play written about him.)
In Julius Caesar, it seems that Antony is set on avenging Caesar death--his funeral speeches in Act Three, scene two seem to point towards that. However, at the beginning of Act Four, Antony is making a hit list with Octavius and Lepidus and wants to find a way to deny the commoners the money Caesar had left them in his will. Antony also speaks badly about Lepidus when he leave to do an errand for Antony. It seems as though Antony's motives are less noble, and are now more about power and success.
In Act Five, especially at the beginning, Octavius is beginning to challenge Antony. Octavius is also being called "Caesar"--a sign that he is coming into more power, and perhaps, taking over what Antony thought was his. At the end of the play, it is Octavius who decides what is to be done with Brutus.
Though it seems that Antony has succeeded in avenging Caesar's death (buy securing the deaths of those who plotted to kill him), Antony didn't achieve as much success as perhaps he was hoping to.