I'd say Shakespeare was not really advocating democracy in this play, but he wasn't too keen on monarchy or a Senate either.
Brutus joins the conspiracy because he believes (due to the planted forged letters and maybe his ego) that the people would rather not be ruled by Caesar. HE believes, in a sense, in democratic, or at least representative republic rule. Showing that Brutus can be duped into such extreme measures illustrates the flaws in a representative government. Cassius manipulates him too easily.
Caesar, on the other hand, actually is flawed in the way the conspirators believe; his eagerness to go to Senate to accept a crown despite the warnings and his lording over the Senate hints at his corruption.
But rule by the people is flawed as well--Note how the crowd is so easily swayed by any strong speaker at the funeral. First they support Pompey, then Caesar--then they back the conspirators--then just as quickly change to follow Antony. When left to their own devices, they riot in the streets and kill innocent poets. Certainly Shakespeare is painting the common people as unable to lead themselves. They are too fickle and easily manipulated and also too impulsive for their own good.
So even though I don't see Shakespeare advocating any OTHER form of government actively--I don't see any support for democratic rule. . They don’t think for themselves. They are incapable of ruling themselves wisely.
There is definitely concern about the "mob mentality" of crowds in Julius Caesar. For instance, the poet Cinna, on his way to Caesar's funeral, is attacked and killed by a hysterical mob.
Moreover, Mark Anthony's famous "Friends, Romans, countrymen," speech is, as critic Mark Bold argues, a rhetorical masterpiece of crowd manipulation. As Brutus has gained the support of the Citizens for the conspirators' murder of Caesar, so Anthony's brilliant speech unwinds this approval and unleashes the hysterical hatred of the crowd against the plotters...(T)he insistent reassurance that Brutus 'is an honourable man' is devastatingly ironic."
Given the gullability and tendancy of large groups toward reprehensible behavior, it is hard to fathom a workable democracy.