If you want a brief answer, the "sullied flesh" Hamlet refers to is his own body: he means it is dirty or tainted in some way.
To expand on that a bit, Hamlet is saying that his body, his physical self, is so rotten in some way that he wishes it would just melt away or evaporate. He's feeling so bad after his father's death that he just wants to die or not be.
Now, the idea of being sullied relates to a more general sense of corruption that runs through the entire play. Remember the line "something in rotten in the state of Denmark"? The entire country is sullied--or tainted--by the murder of Hamlet's father, and by having the uncle--the murderer--now running the country.
If you look at the rest of the monologue in which this phrase appears, you'll see an ongoing reference to things that are rotten: "rank," "gross," stale," etc.
While some texts have the word "sullied" at this place, others have the word "solid." It seems more likely to me that Hamlet would be saying, "O, that this too too solid flesh would melt." It cannot melt because it is too solid. My text is published by Yale University Press and subtitled "The Annotated Shakespeare." The word "solid" is annotated simply with "some texts have 'sullied.'" I like "solid" better because it does not require any explication. It is self-evident. Whereas "sullied" is extremely difficult to interpret, although it would seem to suggest that Hamlet feels tainted by the marriage of his mother to his uncle, or perhaps that all human flesh is tainted by sin.
I note that in the etext provided by eNotes the line is rendered:
O, that this too too sullied flesh would melt
But in the Modern Translation right next to it the line is rendered:
O, that my too, too solid body would melt