How does Shakespeare's portrayal of Julius Caesar differ from the historical figure?

On the whole, you'd have to say that Caesar comes across as a much weaker character in the play than he was in real life. But there's a reason for this; Shakespeare wants to make him a more vulnerable, more sympathetic character. If he were portrayed as every bit as ruthless and ambitious as he was in real life, then, when it came to the bloody Ides of March scene, the audience might well have been cheering on the assassins as they plunged their daggers in.

Although Caesar dominates the action of the play—even though he's not on stage for very long—he's deliberately under-drawn, to the extent that the other characters in the play can pretty much make of him what they will. We get to know Cassius and Brutus's Caesar, the would-be tyrant who wants to make himself king; we also get to know Caesar the selfless servant of the people, the man who, according to Mark Antony, only wanted to do what was best for Rome.

But what we don't get is a rounded historical portrait of Caesar, the kind that would leap straight out of the history books. This allows the other characters to take center stage, as Shakespeare always intended, making the bitter political struggle between them the focal point of the drama.

William Shakespeare's Julius Caesar is based on historical events that occurred some 1600 years earlier. Sources such as Suetonius and Plutarch, writing a century after Caesar's death in 44 BCE, also tell us of a soothsayer who told Caesar to beware of the Ides of March; that Caesar's wife Calpurnia had a terrible dream about her husband being killed; and that Caesar did make some comment to Brutus during the assassination. The ancient sources also tell us that Antony did try to put a crown upon Caesar's head and that he refused the crown. Ancient sources also anticipate Shakespeare in saying that Caesar had epilepsy:

Suetonius tells us that "He was twice attacked by the falling sickness during his campaigns" (Thomas Browne translator), although he says nothing about the attack of this disease that Shakespeare mentions in his play. 

Interestingly, Suetonius tells us that several of the assassins were designated in Caesar's will as "guardians of his son, in case one should be born to him" and that Caesar also named in his will "Decimus Brutus even among his heirs in the second degree" (Thomas Browne translator). Antony omits mention of these things in Shakespeare's version of the play. Shakespeare does follow Suetonius, however, in mentioning that Caesar left some of his gardens and a gift of money to each Roman citizen:

To the people he left his gardens near the Tiber for their common use and three hundred sesterces to each man. (Thomas Browne translator)

So, for the most part, Shakespeare does seem to follow the historical sources to a signficant degree. The playwright does, however, seem to make Caesar look like a more sympathetic figure and downplay Caesar's ambition, especially in the funeral speech given by Antony. Suetonius does suggest that certain rumors about  Caesar's future plans led the conspirators to act quickly:

....the report had spread in various quarters that he intended to move to Ilium or Alexandria, taking with him the resources of the state, draining Italy by levies, and leaving the charge of the city to his friends; also that at the next meeting of the Senate Lucius Cotta would announce...that inasmuch as it was written in the books of fate that the Parthians could be conquered only by a king, Caesar should be given that title. (Thomas Browne translator)

How does Shakespeare's Julius Caesar differ from actual historical facts?

Julius Caesar was almost certainly based upon Plutarch's Lives, a compendium of biographies published in the first century CE and translated into English during Shakespeare's life. Much of the plot of Julius Caesar, and even the lines attributed to certain characters, are derived more or less straight from Plutarch. So, in many ways, the question is how true Plutarch was to facts—and this is very difficult to ascertain, given that Plutarch is one of the most important sources for the life of Julius Caesar as a historical figure. That said, one major deviation from the historical record is that Shakespeare collapsed nearly three years of events, from Caesar's assassination to the battle of Philippi, into a few days, for the purpose of maintaining the flow of the play. Shakespeare also emphasizes the virtues of Brutus, relative to the other conspirators, more than Plutarch does. Even Caesar himself comes off as a more sympathetic character in the play than history might allow—there is talk that Caesar is "ambitious" but little detail about the political (and military) maneuvering that he undertook to put himself in the position to effectively rule over Rome.