One option might be to turn to real-life fortune-teller-esque techniques. Essentially, psychics use these to demonstrate their great insight to the audience. While these could be useful tools for convincing others that you can see the future, you might also use them as they were originally intended for some flavor. I've included the Wikipedia links, because they do an excellent job of explaining and expanding upon them.
Cold reading is tricking your audience into thinking you have powers based on general psychological knowledge. This boils down to making broad statements, likely about a large group of people/things, such that the statement will be true for at least one of them. For example, your PC might say, "I sense impending misfortune. One of you will fall sick in a short time," to a group of people. In almost all cases, this will be correct. Alternately, you can predict something common, but notable, about a single person's future. You might tell a shopkeeper that they will have a fight with someone in their family, or tell an adventurer that they will be injured. In general, keep your statements vague,
Hot reading refers to using prior knowledge, gained without the audience's knowledge, to pretend that you have special insight. This might be more mechanical; you might secretly make Listen checks to overhear conversations, Sense Motive checks to determine truth from lies, or Gather Information checks to find out more about someone, or, of course, just read their mind via magic. Then, you can present it as if you used psychic powers to determine it. You might use these skills to determine information, then interpret it yourself, or ask your DM to let you roll to use your character's Wisdom to interpret it.
For example, you might ask around about the Lord of Greenwood, find out that he loves to fight, and that he will hold a tournament next month. Then, approach him, presenting yourself as a stranger with no knowledge of the area, and "sense" that he'll soon become greatly admired for his skill. This statement stays vague (after all, you'd be in trouble if you predicted that he'd win and he lost in the first round!) but is still specific to him, and he will take it positively.
To read someone's personality, make vague, positive statements about them. These are commonly known as Barnum Statements, and is sometimes called warm reading. If you make these statements contradictory, in what's known as the rainbow ruse, they will always apply. For example, if you say, "You are a warm and kind person, but sometimes you're unable to show how you feel," it will appear that you have deep insight about them, but the statement applies to just about everyone.
Overall, remember to keep your statements vague and open-ended, and allow the people you tell the fortunes of to interpret your words themselves. If you want to keep the fact that you're not really a fortune-teller secret from your fellow players, pass notes with the DM, even if they're actually blank. That way, it looks like you're using your powers, when you're really just using simple ways to make statements apply anywhere.