When Mercutio says, "Ask for me tomorrow, and you shall find me a grave man," he is indeed making a pun on the word grave, which means both serious and a hole in which a dead body is buried. I think such a pun, usually a comedic device, illuminates his character even further because it comes at a markedly not comedic moment in the play. Mercutio has just been stabbed and knows that he is about to die, and yet, for some reason, he is still making jokes. 

All along, it seems that Mercutio has been unable to deal with strong feelings, either in others or in himself. When Romeo is depressed because Rosaline does not return his love, he doesn't want to go to the Capulets' party and have fun because feels as though he has a "soul of lead." Mercutio lectures Romeo on the nature of dreams, saying that they are "the children of an idle brain, / Begot of nothing but vain fantasy." He cannot recognize the strength of Romeo's feelings.

It happens again when Romeo runs off into the night to look for Juliet after the party. Mercutio calls for him, mocking him the entire time: making light of love in general and Romeo's love (of Rosaline) in particular, even using very sexual and bawdy talk. Mercutio makes love all about sex, failing to recognize the strength of his friend's feelings for Rosaline (or Juliet). In fact, he makes so many sexual jokes during the play that he begins to seem incredibly immature, emotionally speaking. I think we can look to his "grave" pun in act 3 as further confirmation of this: he has failed to take responsibility for his own part in the violence with Tybalt, he makes jokes, and then he blames the two families for his death, which was, in many ways, his own fault.

Mercutio utters this famous line as he is stumbling around, following his stabbing by Tybalt.  At this point in the scene, the others are unaware that Mercutio has been injured to the degree that he has; Mercutio has been playing this up as more of a "flesh wound" (as Monty Python's Black Knight would say), and the others buy into his joking.  In fact, just before Mercutio utters this line, Romeo encourages him by saying, "Courage, man.  The hurt cannot be much!"

In reality, the hurt is not only "much," it is fatal.  Mercutio will fall down shortly afterward and we will learn of his death soon after.  Thus, the foreshadowing of this particular line.  On the surface, we can see that Mercutio intends "grave" to mean "serious" -- ask for me tomorrow, and things might be serious. However, in the other definition of "grave" -- that of a final resting place for the dead -- we see Mercurio's true intention for the statement.  He is realizing the wound is fatal and that he will not live until tomorrow.

Mercutio is making a pun with his tragic wound. During the fight between he and Tybalt, Romeo comes between the two of them and Mercutio is stabbed by Tybalt, which was not his intention. Tybalt is encouraged by his kinsmen to leave quickly and Mercutio goes into his final speech. Mercutio is dying and knows he will soon be in the "grave." He continues to make light of the situation and everyone feels he is just "clowning around" as always and they do not take him seriously. Even when he falls down the steps in the town square and is dead they think he is just fooling around till Romeo lifts the handkerchief and sees the wound in his heart.

Mercutio has just been mortally wounded by Tybalt and knows he is dying. In trying to cover it up, he is still making jokes. His pun is appropriate since he will be dead and in his "grave" tomorrow. The others think he's referring to the meaning of grave as "serious".

Level-headed, sensible, non-romantic Mercutio is engaging in a bit of word play here in Act 3.1,98-99. He knows that he will soon be dead from the wounds inflicted by Tybalt and will indeed be in an actual grave.

The fact that Mercutio has lost his life for someone else's cause ("A plague on both your houses" 100-101) and at the hands of someone whom he considers inferior is doubly ironic for the character ("Zounds, a dog, a rat, a mouse,/a cat, to scratch a man to death" 101-102).