According to the American Heritage Dictionary, the word canker can be defined as "a source of spreading corruption or decay."
To fully understand this description of Claudius, you need to look at in the context of the description and the play as a whole.
Hamlet has just shown Horatio the letter from Claudius calling for his (Hamlet's) execution by England. When Horatio questions what sort of king would do such an act, Hamlet replies with the following:
"Does it not, think thee, stand me now upon-
He that hath killed my king, and whored my mother;
Popped in between th' election and my hopes,
Thrown out his angle for my proper life,
And with such cozenage-is't not perfect conscience
To quit him with this arm? And is't not to be damned
To let this canker of our nature come
In further evil?"
In essence Hamlet is listing a few of Claudius' wrongs--his corruption, if you will. Namely, Claudius has:
1. Killed a king
2. Made a whore of Hamlet's mother
3. Taken Hamlet's crown
4. Attempted to have Hamlet murdered
Hamlet then goes on to tell Horatio that his conscience compels him to "quit him with this arm," meaning kill Claudius with his own hands. He goes on to say that he deserves damnation if he allows Claudius, through his own inaction, to spread his corruption by committing another evil act.