What are the internal and external conflicts of Caesar in the play Julius Caesar?

Often, many readers examine the internal and external conflicts of Shakespeare's Julius Caesar in relation to Brutus, but it's important to realize that Caesar himself faces internal and external conflict (though he has precious little time on the stage before his untimely demise). In a nutshell, the internal conflict Caesar faces is his own paranoia, while the external conflict he faces is the murderous conspiracy of dissatisfied Romans. 

While Caesar begins the play as the most powerful man in Rome, one can also surmise that he's paranoid. When considering potential advisors, Caesar proclaims:

Let me have men about me that are fat;

Sleek-headed men, and such as sleep o' nights.

Yond Cassius has a lean and hungry look;

He thinks too much. Such men are dangerous. (192-95)

In this quote, we see that Caesar prefers to be surrounded by advisors who don't think. As such, one can surmise that he already fears potential threats to his authority, and so he wants to avoid people intelligent enough to oppose him. This preference subtly illustrates the paranoia Caesar internally struggles with.

The external conflict Caesar faces is of course more obvious: it's the conspirators. Backed by helpers, Cassius, and later Brutus, plot to wrestle power away from Caesar and protect the Republic. The conflict is relatively short and very violent, as the conspirators ultimately succeed in overcoming and killing Caesar in Act 3, Scene 1.