So as no flaired user has answered this, and because I think the other answer in this thread isn't really specific to anything Shakespearean, I'm going to take a stab at this.

As ever with answers on this sub, I think it's important to state from the outset that how you conceive of Elizabethan and early Jacobian theatre is probably very different to how it actually was. For most plays, most of the time, it was not the case that the audience were, as in contemporary theatres, seated in basically identical seats (barring boxes) that are only price-differentiated by their location and view of the stage. Furthermore, it was not the case that a play was written, performed by a large variety of different companies travelling all across the country in a multitude of theatres - whilst this did happen, Shakespeare himself was the company playwright of the Lord Chamberlain's Men, later the King's Men, and they were the resident company at the Globe Theatre, a theatre on the south bank of the Thames that has been reconstructed pretty accurately. For much of his writing career, Shakespeare was writing plays that were being commissioned by the Crown for the court, and they were being written with a very specific theatre and a specific company in mind. It's worth bearing this in mind, because contemporary playwrights, especially those who aren't attached to a company or don't have a residency, are writing for a much broader purpose than the Bard was.

So let's take a look at the globe. This first image is a wikipedia reconstruction of the original Globe (which burnt down), this second one is a photo of the reconstruction today empty, and this third one is the current Globe full of people.

So what you can see is that there are four tiers, three seated and one standing. Generally, speaking, the lower down you are, the more uncomfortable your position is, it's going to be harder to see the actors' doing various bits of 'high brow' acting (especially when it comes to the more complicated soliloquies), you're more likely to get pissed on by the actors or splashed with whatever props are on stage - pig's blood being a regular - and the cheaper your tickets are going to be. So what does this all mean?

Was the audience mostly aristocrats or the middle class or the poor?

The audience was everybody. Whilst popular conceptions of Early Modern England depict it as a highly stratified society, all kinds of people were going to see shows at the Globe, and there were ticket prices for them all. If you're in the stands, you're probably quite poor, it's unlikely you're literate, and your education is going to be a hodgepodge at best. But what you will enjoy is the visceral action (remember the pig's blood etc.), the lewd jokes, the gestures the cast are making and, of course, even if you can't follow every word or understand every reference, Shakespeare is very good at inserting small expositionary monologues or dialogues that sum up what's happened in an act or scene, so you're getting your money's worth.

As you go higher up the theatre, you're richer and more educated, including high ranking members of the nobility and potentially even some royalty (although it's far, far more likely that the company would go to court to put on a showing of their new play). Clearly, then, these people are going to progressively "get" the higher literary and historical themes more and more in line with their education - those who are literate and have access to literature, especially, will appreciate all the references and would understand that what they were watching was a dramatised retelling of famous historical events, and understand who the characters were.

So the answer to your question as posed doesn't really exist. There is no average theatre-goer in the Elizabethan or Jacobian Globe who we can make sweeping statements about. There is no corps of middle-upper class types, as there are now, who are frequently (and in some cases exclusively) patronising a plethora of theatres regularly. Some members of the audience will be highly educated, wealthy, influential people who will understand virtually everything the cast are saying and understand its context. There are people who will appreciate some of it, and will understand a lot of it, but won't have the educational background to contextualise it in the way that contemporary students of literature do. At the other end of the spectrum, there are people who've paid barely anything for their ticket, and are just there to enjoy the show - whatever it is.