That Cassius of "Julius Caesar" is a tragic hero is questionable. First of all, a tragic hero is a man of noble stature and his fall results from his committing "an act of injustice" (hamartia) either through ignorance or from a conviction that some greater good will be served.
When the reader first meets Cassius, there is no mistaking his lack of noble stature. For, he is excessively proud while at the same time envious of Caesar. He tells Brutus how he has been "born free as Caesar" (I,ii,97), but when he and Caesar swam together, Cassius had to save the ruler from drowning:
But ere we could arrive the point proposed,/Caesar cried 'Help me, Cassius, or I sink!'/I, as Aeneas, our great ancestor,/Did from the flames of Troy upon his shoulder/The old Anchises bear, so from the waves of Tiber/Did I the tired Caesar. And this man/Is now become a god, and Cassius is/A wretched creature, and must bend his body/If Caesar carelessly but nod on him. (I,ii,110-118)
In his persuasive speech with Brutus, the "seduction scene," Cassius's low emotions also are shown as he seeks to sway Brutus to join in the conspiracy. He even stoops to planning to send forged letters to Brutus in order to convince him of the need for Caesar's death:
I will this night /In several hands, in at his windows throw,/As if they came from several citizens,/Writings, all tending to the great opinion. (I,ii,315-318)
The motivation for greater good is certainly suspect here as Cassius plans his underhanded activity. Unlike Brutus, who feels that Caesar's death is for the greater good, Cassius desire for the ruler's death is motivated by his disastifaction with Caesar and by his personal grievances.
Regarding Brutus, Cassius does make mistakes because of fate or hamartia. As the rational man who has said that "the fault of each man is not in the stars, but in ourselves, that we are underlings" (I,ii,140-141), Cassius displays his discontent and personal envy when he says of Caesar,
Why, man, he doth bestride the narrow world/Like a Colossus, and we petty men/Walk under his huge legs and peep about/To find ourselves dishonoraable graves. (I,ii,135-138)
Cassius's mistakes are not made out of ignorance or from believing in a greater good; instead, Cassius's mistakes are made from a love for Brutus. He allows Marc Antony to live after the assassination of Caesar; against his better judgment, as he acquieses to the decision of Brutus to allow Marc Antony to address the crowd. Later, he quarrels with Brutus, and complains that Brutus no longer loves him:
O, I could weep/My spirit from mine eyes! There is my dagger,/And here my naked breast...If that thou be'st a Roman, take it forth. (IV, iii, 98-102)
Finally, in Act IV, Cassius concedes to Brutus in a fatal military battle. It seems that whenever Cassius hears Brutus's logic, his nature twists on itself, and showing his love for his brother-in-law, follows his directives, rather than his logic. A noble hero would not do so. Therefore, rather than carrying out the theme of ambition as Caesar and Brutus do, Cassius and the other disgruntled characters of "Julius Caesar" further the theme of envy and disastifaction.
I couldn't agree more with the excellent discussion in post #2. There is, indeed, nothing heroic in the character of Cassius. Brutus is the tragic hero of Julius Caesar. It is Brutus whose life is destroyed by a fatal flaw within his own character, his idealism that blinds him to the realities and the workings of human nature. Cassius, in fact, functions as the antithesis of Brutus. Brutus is honest; Cassius is devious and deceitful. Brutus is noble and self-sacrificing; Cassius is selfish and base. Brutus loves his wife, his country, and even his friend Caesar; Cassius loves only himself. These differences in the two characters--in fact, the difference between Brutus and every other conspirator--is spelled out in the conclusion of the play, ironically by Antony:
This was the noblest Roman of them all.
All the conspirators, save only he,
Did that they did in envy of great Caesar;
He only, in a general honest thought
And common good to all, made one of them.
One way Cassius contributes to the theme of the play, it seems, is by showing us very clearly what a hero is not.