In Acts Two and Three of Shakespeare's Hamlet, unfortunately, Hamlet does not succeed in avenging his father's death as he had promised Old Hamlet's ghost.
One of the big reasons Hamlet fails is due to his tragic flaw, which is indecision. Hamlet struggles to make up his mind regarding a number of things, primarily when to strike down Claudius, his father's brother and murderer.
I find, however, that I am sympathetic to Hamlet's plight. During the period in which Shakespeare wrote and produced this play, the Elizabethan audience was both religious and superstitious. In terms of religion, they believed that God ordained who should be king, and to kill a king was to offend God. On the other hand, the Elizabethans believed in the Devil and witches that served him. It was their understanding that witches were placed on the earth in service to the "powers of darkness," in order to trick humans unto their eternal damnation.
The third element of this play that is important in all this is the recent marriage between Claudius (Old Hamlet's brother) and Gertrude, Old Hamlet's widow. Elizabethans believed that when two people married, they became one. Basically, they were no longer separate entities. Therefore, when Old Hamlet dies, part of him lives on in Gertrude, and when she marries, the Elizabethans believe Claudius and Gertrude are committing incest because Claudius is actually sleeping with the part of his brother that remains within Gertrude. (Remember this point.)
So Hamlet is caught between a tough choice and an even tougher choice. If the king is not a murderer, Hamlet could go to Hell for killing him. If the ghost that appears to Hamlet is a false ghost and not really his father, Hamlet could, again, forfeit his soul's salvation. It is easy to understand why Hamlet hesitates.
In Act III, scene iii, Hamlet, having seen Claudius' incriminating response to the play that has reenacted his murder of Old Hamlet (in scene ii), knows finally, that Claudius is guilty, but when he comes upon him while praying, Hamlet realizes that if he kills the King, his soul will go to Heaven.
Now might I do it pay, now hi is praying, /And now I'll do 't. And so he goes to Heaven, / And so I am revenged. That would be scanned; / A villain kills my father, and for that / I, his sole son, do this same villain send / To Heaven.
So Hamlet decides to wait. In scene iv, Hamlet is ready to act when the circumstances are right. He visits his mother, while Polonius hides in her room behind a curtain (arras) in order to listen to what Hamlet says. Ironically, Hamlet—believing that Claudius has just been sleeping with Gertrude and is now hidden behind the curtain—thinks that if he kills Claudius at this moment, Claudius will go to Hell with the sin of incest on his soul, and so Hamlet stabs the person behind the curtain; we also know it is the interfering Polonius, and not Claudius, who dies.
Even after all this, Hamlet is still unable to fulfill his promise to Old Hamlet's ghost.
To directly answer your question, Hamlet is completely unsuccessful in avenging his father because he simply doesn't kill Claudius in either Act II or III. But, then the question becomes: does Hamlet's inaction (not killing him) serve his overall purpose - Or, do Hamlet's actions in these scenes avenge him in subtle ways.
In Act III, Scene iii, Hamlet has a chance to kill Claudius but refrains because the King is praying. Hamlet doesn't realize it, but Claudius is just offering words, not thoughts. Hamlet was worried that killing Claudius in this penitent mode would just send Claudius to heaven. Claudius says, "Words without thoughts never to heaven go." - meaning that he was just offering thoughtless prayers.
In Act III, Scene i, Hamlet devises the "mousetrap" to "catch the conscience of the king." Hamlet, in his hesitation and over analyzing, does come to the conclusion that it is not enough to just kill him. He must expose his conscience to the world (court).
Hamlet has also frequently referred to his crafty madness and that could be seen as something just to play with Claudius' mind; to always keep him on edge; a kind of psychological torture as Claudius is never certain what Hamlet's intentions are. Remember that Hamlet is in line for the throne, so that threat is also always there, whether or not Claudius knows Hamlet is aware that he killed is father. So, all the subtle lines, (like Gertrude married hastily), Hamlet's mood, and the play all point toward Hamlet's pscychological game with the King en route to his avenging his father. Hamlet keeps everyone uneasy. In that respect, you could say he is successful.