When the audience first meet Leontes at the beginning of Act 1 Scene 2, he seems to show very little sign of jealousy towards Polixenes, referring to him as 'brother' but by the time of his exit on line 350 has asked Camillo to 'give mine enemy a lasting wink'
As the scene opens, Leontes tries to convince Polixenes to stay in Sicilia longer, 'One sev'night longer' and the way he repeats this suggests that he is sincere in asking Polixenes to stay, rather than simply being courteous. When Polixenes refuses his request and says he will leave 'tomorrow' Leontes again presses him to stay saying they should compromise and 'part the time between' them. At this point Leontes seems neither hostile towards Polixenes nor suspicious of a relationship between him and Hermione and, a few lines later invites Hermione to speak, 'Tongue tied our queen? Speak you.' Leontes invites Hermione to speak freely to Polixenes and asks her to help him convince Polixenes to stay.
It seems very unlikely that Leontes would have deliberately encouraged them to speak freely or be alone together, as he leaves the stage a few lines later, if he suspected them to be having an adulterous relationship.
However, Leontes could easily have already been suspicious of the relationship between Polixenes and Hermione and, after Polixenes not agreeing to stay for him, be testing to see if he will agree for Hermione.
When Leontes reappears on stage after Hermione and Polixenes conversation he immediately asks his wife, 'Is he won yet?' and, on Hermione's reply of, 'he'll stay, my lord' remarks, 'At my request he would not'. This is the first obvious sign that Leontes is jealous of Hermione's relationship with Polixenes, but he quickly rectifies his bitterness, saying, 'Hermione, my dearest, thou never spok'st to better purpose'. This could easily be read as Leontes being suddenly very nice to Hermione, attempting to compensate for his earlier remark. This could show that Leontes has been suspicious of a relationship between them for some time but has been carefully hiding it until that point.
If Leontes asking for Hermione's help in persuading Polixenes to stay in Sicilia longer had merely been a test to confirm his suspicious of a relationship between them, then the fact that Polixenes did agree to stay for Hermione would have convinced Leontes he had been cuckolded. With his suspicions confirmed, it would have been safe for him to express his jealousy, knowing it was supported.
This is also shown in that, up until this point, and his following aside, Leontes has only spoken in very short lines since the beginning of the scene. But, the length of his speeches increase as he becomes more jealous. This could indicate that, although he was suspicious at the start of the scene, he was trying to hide his feelings, as now feels no real need to.
And in his following aside he becomes suspicious, describing the possible actions
There is a lot of punctuation used in the middle of the lines of Leontes aside and following speeches, 'My bosom likes not, nor my brows - Mamillius, art thou my boy?' These break the natural speech pattern and represent Leontes' disjointed thoughts, as he becomes seemingly irrationally jealous.
As he describes the actions of Polixenes and Hermione, he uses alliteration, 'paddling palms and pinching fingers... practised smiles', the short, sharp 'p' sounds mirroring his bitterness and jealousy.
He describes Polixenes and Hermione's actions as being quite intimate, 'paddling palms'. It is very unlikely that they would actually be present on stage holding hands in such an intimate way with Leontes present, even if they were in a relationship, as Leontes suspects. He seems to exaggerate things he is seeing and imagine every action to be intimate or sexual which may indicate that he is merely paranoid about his wife's relationship with Polixenes.