Incidentally I was interested in this question a couple of months ago, may be these remarks will be useful.
1) PSL(2,7) remarkably isomorphic (MO 37525) to GL(3,2). (See also Wikipedia page and Vipul Naik's site).
2) It is simple group so all irreps are faithful.
3) There are several on-line expositions around irreps of GL(2,F_q), SL(2,F_q). Not sure PSL(2,F_q) is there, but nevetheless might be useful to give these links:
Paul Garrett: http://www.math.umn.edu/~garrett/m/v/toy_GL2.pdf
Etingof&Students: http://arxiv.org/abs/0901.0827
Amritanshu Prasad http://www.imsc.res.in/~amri/html_notes/notes.html#notesch2.html
(This also discusses GL(n,F_q) partly).
Also the book by Fulton Harris discusses the GL(2,F_q) irreps.
4) The group PSL(2,F_q) has irrep of dimension "q" which actually can realized over reals (actually rationals). It can be seen like this: PSL(2,F_q) acts on the projective line P^1(q). This is a set of q+1 points. So we have a q+1 represenation realized in functions on this set - as any permutation representation it contains the constants as 1d irrep. The complement to constants will give you q+1 dim. irrep.
6) For the particular case of PSL(2,7) we may try to construct irreps inducing from the Sylow's cyclic subgroups of orders 7 and 3. You might enjoy M.Isaacs MO answer on my question about decomposition of these irreps.
7) These were over mainly about complex numbers irreps. I cannot say much about irreps over reals, but here are some remarks. There are 6 irreps over complex numbers (I have inserted character table to Wikipedia page) dimensions 1, 3,3, 6,7,8. Actually 1,6,7,8 can be realized over reals. About 7 - see discussion above, 1d - is trivial about 6,8 - let me refer to this page which calculates Frobenius-Schur indicators and shows that they are equal to 1 (hence irreps realized over reals) for 1,6,7,8.
Now there are two 3d irreps (cuspidal irreps) they are complex conjugate to each other, so their direct sum can be realized over reals.