What are Hamlet's real feelings for his mother?
In Shakespeare's Hamlet, Hamlet is obsessed with his mother.
Even though the Ghost tells him to not
...let thy soul contrive
Against thy mother aught. Leave her to heaven,
And to those thorns that in her bosom lodge
To prick and sting her... (Act 1.5.85-88)
he cannot do so. Often, his mother's remarriage seems to bother him more than his father's murder.
Furthermore, he's obsessed with Gertrude's sexual relations with Claudius. In Act 3.4, after he's mistakenly killed Polonius (thinking it's Claudius), and after the Ghost has reappeared to him to urge Hamlet to not forget his "almost blunted purpose" and to tell him to comfort Gertrude, her sexual relations still dominate his words. He instructs his mother to:
...live the purer with the other half.
Good night--but go not to my uncle's bed.
Assume a virture, if you have it not.
And he even gives her specific advice on how to go about achieving this abstinence:
And that shall lend a kind of easiness
To the next abstinence; the next more easy;
For use almost can change the stamp of nature,
And either curb the devil, or throw him out
With wondrous potency.
And when Gertrude specifically asks him what she should do, although he ultimately tells her to tell Claudius that he is mad so that Claudius does not know he is faking, he prefaces this with:
Not this, by no means, that I bid you do:
Let the bloat king tempt you again to bed,
Pinch wanton on your cheek, call you his mouse,
And let him, for a pair of reechy kisses,
Or paddling in your neck with his damned fingers,
Make you to ravel all this matter out,...
This is obsession, not just normal love or concern or hurt feelings. Some suggest it is unnatural, and some performers perform it as such.
Incidentally, you probably don't need the word "real" in your question. Hamlet's disgust for what Gertrude has done, his hurt, his disdain, his condescension, his obsession, as well as his love, are all quite apparent. His words and actions to and toward his mother are quite revealing to the reader/viewer, even if Gertrude doesn't understand.
Hamlet's feelings toward his mother are complicated and conflicted and move from sympathy to hatred and back to sympathy through the course of the play. When he learns that his father has died and that Gertrude has married her dead husband's brother, Hamlet is concerned. After he sees his father's ghost, he is convinced that Claudius murdered the king, and he suspects his mother's involvement. He is angry, horrified, and even somewhat jealous of his mother. His anger wins out and he accuses his mother of complicity in his father's murder, at the same time, accidentally killing Polonius, who had been eavesdropping on their conversation. He does not see how she can so easily switch her attentions from his father to his uncle, but Gertrude herself is a complicated character. She, like her son, is ambitious, and she knows that a woman alone or a woman with a son on the throne is fair game. Hamlet loves and admires her, but his love cannot hold in the face of the murder of his father, and he cannot really forgive his mother for what she has done.