I am writing a letter from Banquo (who is dead) to Fleance about his suspicions about Macbeth. What were Banquo's suspicions?

Once Banquo has been killed by the three murderers hired by Macbeth, it is likely that he would suspect Macbeth of the deed. Banquo has previously shown some suspicion of Macbeth, noting his odd behavior when he received the prophecies from the Witches and when he was wandering the castle grounds late the night of Duncan's murder. Since Banquo was with Macbeth to hear the Witches' predictions, he knows that Macbeth was supposed to become king one day, and he also knows that Macbeth has heard he (Banquo) will be father to kings. Banquo and Macbeth are longtime friends, and it is possible that Banquo senses Macbeth's jealousy of Banquo's own prophecy. The fact that the murderers also tried to kill Fleance while he was with Banquo would further support that motive and Banquo's suspicions.
If we believe Banquo's ghost was consciously at the banquet hall and was able to see Macbeth's reaction to his appearance, that would be further proof of Macbeth's guilt. Macbeth also spoke some lines in that scene that almost directly give away his part in the murder.
In your letter, you could have Banquo relay the prophecies to his son and discuss Macbeth's reactions. You could then talk about the banquet scene in Act III when Banquo's ghost appears and use details about Macbeth's reaction to the ghost that would further support his guilt and Banquo's suspicions. Surely Banquo would also warn his son to flee Scotland until Macbeth is no longer in power.