That I, the son of a dear father murdered,
Prompted to my revenge by heaven and hell,
Must, like a whore unpack my heart with words
And fall a-cursing like a very drab,
A scullion!" (2.2.611-616)
This sentence is basically the tipping of the iceberg to Hamlet deciding he will kill Claudius. He is fed up with simply speaking his discontent in the scenario he is forced to live with, and sends out his players to kill Claudius and gain vengeance instead of being all-talk. He realizes that he should be taking action for what has been done, almost pumping himself up in this sentence by saying he doesn't want to be a scullion (a servant).
This is important to the play because it is this epiphany that will ultimately depict that fate of not only Hamlet and Claudius, but also the state of Denmark. In general terms, attempting to kill anyone of high power will wreak havoc, and now that Hamlet has set his ambitions to do so with Claudius, it can only result in chaos and tragedy--both of which are Shakespeare's fortes apparently.
Hamlet now is taking that turn into the second year of the Miami Heat, and is gearing up towards gaining that championship-in his case, revenge for his father's death.