A key thing to comprehend, with regard to playing your character, is that you need to get into your character's head and perceive the environment from his perspective. This includes the experience of using your character's abilities, so you need to imagine the physicality he exhibits when he uses certain abilities. Once you can do that, you will find that your perception of how your man thinks will become far more lifelike and therefore your performance will greatly improve.
While this is applicable to any character in any role-playing game, for a psychic it is very significant due to the interaction of his powers with his body and his environment. The game reflects that the use of psychic powers taxes the user's endurance by the use of Inner Strength Points, which means that the usage of psychic powers is very much a usage of power channeled through the user's body- something that is finite within a given interval of time, but extendable with experience as it reflects conditioning gained as a byproduct of earning that experience.
So, take a moment to imagine how your psychic experiences his powers. He's always had them, even if they didn't manifest until his adolescence or adulthood, so there was always some telltale sign of their presence; he heard things, saw things, felt things, etc. that others did not--or, at least, others not psychic--and these signs grew until his latent powers manifested and blossomed into full expression. Depending on what powers you choose, your character's early life and experiences with his powers just about tell you the story for you because they won't happen any other way.
On a related note, take a moment to imagine what it feels like when your psychic uses his powers. From some point in his brain, a thought resonates at just the right frequency to become the right effect. He will instinctively use his body, as necessary, to guide the use of this power and employ it to the desired end; a Mind Melter or Psi Warrior will, without thought, form one hand into a firm grip upon the Psi Sword that he wills into existence while a Burster will scan with his eyes the place where he'll ignite a fire a moment before willing it to spark and flare into existing. Telekinetics will gesture with hands or similar limbs as if they were either manipulating it by touch or conducting it into place. While most of these instinctual gestures are actually not needed (and therefore can be trained out of practice), the psychic's experience of the flow of power from his brain, through his body, and then out into the environment to create the effect is often unchecked by training--Mystics actively avoid it, in harmony with their own belief in emergence and intuitive cognition, for example--often correlates bodily movement (great and small alike) to effective use of power, making psychics and magic-users seem to overlap far more than is actually the case.
You will know a learned, trained, and veteran psychic (therefore) by how often he omits such tells in the use of his powers, but this means that he has to somehow deal with the fact that the instinctual displays also naturally discharge any excess energy (and I mean that in the real-world sense, not a game mechanic sense) that the use of such powers naturally produces. Psychics who are telekinetics (or other psychokinetics, such as Bursters) flail their arms about because it is a very natural method to direct and expel such energies; psychics who deal in their powers otherwise have to find other manners to cope with discharging the adrenal rush that comes with using those powers.
A savvy observer can learn how to read body language to find these tells, and other psychics are very good at learning to do this quicky due to their own psychic ability, making psychics most effective at sensing their own- this is one of the reasons for why Psi Stalkers and Dog Boys are so very effective. There are some natural tells that can't be avoided (usually associated with powers like Psi Sword/Shield and those like it), but most--across all categories--can be omitted with discipline and practice, and replaced with far more subtle ones that allow them to escape the notice of observers. Psychics with a past in organizations (great or small) whose presence persists will do this with their members; this, by the way, is one of the many reasons for why the Coalition States' inner elite of telepaths and other psychics remains well-hidden. (More on this later in the series.)
So, get into your psychic's head. Get into his body. Imagine how it feels to use those powers, and you will go a long way towards comprehending your man's mind and perspective- which will, in turn, make playing your man a far more satisfying play experience.