Of the two characters, Gertrude would seem to be the one we can sympathize with. Once the seed of Claudius being a murderer is planted, it is hard not to see him in that light.
The question becomes just how much does Gertrude know or suspect? Is she innocent? Does she marry Claudius out of love? Does she marry him because she is weak and doesn't know what else to do? Shakespeare doesn't answer these question but leaves it up to the actress playing the part. As for reading the play, there is nothing to indicate that she is involved.
As for Claudius, as I stated, once the seed of guilt is planted by the ghost , it is hard not to see him as guilty. His public face is one of a concerned step father but like Hamlet, we see through the act. It is hard to have any sympathy for him,
I would actually tend to suggest that we should feel very little sympathy for the pair, given that they have just celebrated their overly quick marriage and appear to be enjoying the very wonderful fruits of the murder of Hamlet Sr. Of course we do have some inkling that Gertrude might not be completely comfortable with the whole situation, but Claudius is in hog heaven.
There are also some chinks in the armor as Hamlet talks about being too much in the sun, there are things that he is clearly worried about. But none of these lead us to feel a whole lot of sympathy for the newly married couple.