The way to play this depends heavily on what you're trying to achieve with your antagonist's secret goal. The key concept here is that every major thread should have some set-up and introduction in the first act; how precisely to accomplish that for the "antagonist's goal" thread depends on the specifics of your story.
So, what are your goals for this thread? What kind of story do you mean to tell; what effect are you aiming to accomplish with it? I'll give three likely examples; if these don't cover your specific case, I hope they will at least demonstrate the considerations you can use and how to deal with them.
- Suspense. Your major goal for this arc is for the reader to be aware of a developing impending threat to the protagonist. As the story proceeds, the antagonist makes things harder and harder for the hero - the hero doesn't know that yet, but the reader does, and he's anticipating the climactic eruption and showdown.
- Surprise. This arc is a gradual construction of a major twist which will shock and surprise both the protagonist and the reader when it comes to light. The trusted helper is dramatically revealed to be a traitor. The effect you're going for is a build-up to an "OH MY GOD" moment of surprise from the reader, akin to the legendary "Luke, I am your father!".
- Character. The antagonist is a fully-fleshed character you want the reader to be interested in and have some sympathy for. Basically, you're giving the antagonist a story arc of his own, focused on him and his character, and putting that arc on a direct collision course with the antagonist.
Your own arc will probably be more detailed; intertwined with other story arcs; maybe you'll have multiple goals for the same plot and character. But just as you've described your problem as being with "the antagonist's secret goal," you can figure out roughly what you're trying to achieve by setting him that goal. That's the arc you want to focus on now.
Once you know what your arc is, that's what you need to set up in the first act. In many cases, this won't necessarily be a character goal, or it might be something else in addition to a character goal. So let's see what we need in the first act to make each of our arcs work well:
- Suspense. The suspense arc is based on establishing an impending threat to the protagonist. So the first act needs to introduce that threat. Here, introducing the antagonist's goal in the first act is highly appropriate - you're showing us the protagonist has a threat to be reckoned with, one he won't see coming. That's exactly what sets up the suspense.
- Surprise. If you're aiming for the antagonist's treachery to surprise the reader, then obviously that's not something you want to give away in Act I! Instead, what you want to do is set up the Shocking Revelation arc - lay the groundwork, ease in the foreshadowing. The first act of a surprise arc needs to establish a lot: the expected norm (that the surprise shatters); the first clues that something is amiss (this prepares us for a surprise later on); some of the elements that will (later on) explain the antagonist's treachery. If this is your goal, then you want to keep your antagonist's true aims a secret until the dramatic reveal. Instead, use the first act to subtly set up all the major elements you'll need later on for the surprise to be clear, interesting, and effective. In this case, those (and not the antagonist goal) are the elements that need to be introduced in the first act - otherwise they'll feel disconnected and out-of-place in the later acts, when you'll need them.
- Character. If you're giving the antagonist a whole arc, then that arc will have its own structure. His goal of framing your protagonist might be central to that arc (in which case, you'll definitely want to get it in early on, in Act I); or it might be a means to a far more important end for him (for example, maybe his actual goal is helping his sick sister, and framing the protagonist is a means to that end that develops later in the story - in this case, the goal "help my sister" is what you want to have in Act I; that's what defines his behavior for the rest of the book).
Things can get a little more complex than this - for example, maybe you're planning to start out with a surprise, and then milk it for suspense for a while; maybe you want a character arc and for it to be suspenseful; etc. etc. But the core principle is the same: figure out what you're aiming to do; Act I sets it up, Act II develops and complicates it; Act III is the showdown and climax.
If this answer sounds helpful to you, and you'd like more thoughts on how it might apply to your specific story, feel free to flag me down in the chat room and we can mull it over ;)