Analyze the characters of Leontes and Hermione in Act I and Act II.
Even limiting discussion to Acts One and Two, this is quite a topic, so I will keep my answer more general.
First, Leontes fits into the same category that other protagonists in Shakespeare's plays fit into, which is to say that they share their thoughts, through soliloquies, directly with the audience. Other examples of this are Hamlet, Richard III, Iago, Macbeth, etc. This is important to notice because we, the audience, get a certain insight into these characters' thought processes that gives us some of the reasoning behind their actions.
For Hermione, this is not true. What we know of her and her actions can only be gleaned from her behaviour and conversation with other characters.
Leontes is, from almost the first moment he is onstage, consumed with jealousy. There doesn't seem to be any evidence for this, but that doesn't stop him from weaving more and more fantastical and far-fetched reasons to believe what he does. I think that it is genius on Shakespeare's part to make Leontes' behaviour so outlandish and unexplainable. For, unfortunately, that is often the case in human life when it comes to jealousy.
As for Hermione's behaviour: She spends most of Acts One and Two patiently playing the hostess to Leontes friend Polixenes (with whom Leontes has cast her as an adulterer) and caring for he son. She does these things even though she is very visibly pregnant, a state that couldn't make these duties very enjoyable. Yet she remains pleasant and dignified. Though Hermione potentially could be very unstable emotionally (all those raging hormones!) she is even-tempered and calm.
An interesting note about Leontes is that, though he goes on and on, listing reason and explanation to support his ludicrous claim of adultery between Hermione and Polixines, Shakespeare is able to make it evident (through how erratic and choppy the rhythm of the verse that Leontes speaks is) that this man is not in his right mind.
In contrast, Hermione's speech is even, balanced and clear, even when she is accused out of the blue. She stands up for herself with dignity and grace.