The character who speaks these lines is Cassius, Brutus's co-conspirator (and, indeed, the one who persuaded Brutus to get involved in the first place.) The line is in Act V, Scene 3, where Cassius has just learned that his forces are surrounded by those of Mark Antony. He knows his death is near, and he is struck by the symbolically significant fact that the day is his birthday. So he will breathe his last on the day he first breathed. This causes him to reflect on life as a cycle, or a "compass," which he explains in the next two lines:
And where I did begin, there shall I end;
My life is run his compass.
With this, he has his servant Pindarus run him through with the very sword that stabbed Caesar, yet another sense in which things have come full circle.