The only simple groups having an irreducible character of degree 3 are $A_5$ and $PSL_2(7)$. What is it know for degrees 4 or 5? That is, do we know what simple groups posses an irreducible charcter of degree 4? of degree 5?
I think the best modern source for this information is the paper:
G. Hiss and G. Malle. Low-dimensional representations of quasi-simple groups. LMS J. Comput. Math. 4 (2001), 22-63. Corrigenda: LMS J. Comput. Math. 5 (2002), 95-126.
This lists all irreducible representations (in characteristic 0 and coprime to the defining characteristic of the group, if any) of all finite quasisimple groups (perfect groups with $G/Z(G)$ simple) up to dimension 250.
You can consult the ATLAS to see that the sporadic groups don't figure here.
For groups of Lie type you should refer to the following paper:
There are actually a couple of errors in the main theorem of this paper, and a correct version is given as Table 5.3.A. of Kleidman and Liebeck's The subgroup structure of the finite classical groups.
Finally the alternating groups are dealt with in Proposition 5.3.7 of Kleidman and Liebeck; in particular you'll see that only $A_5, A_6$ and $A_7$ are relevant here.
Note that all of these reults give projective representations; these will all lift to representations of a quasisimple cover of the relevant simple group. To check whether this quasisimple cover is actually the simple group in question you can use the ATLAS.
Note that I have e-copies of all the sources I've referenced and will email you copies if you need them.
Update: Out of interest, I went through the sources above to see which groups have nontrivial irreducible reps of degree at most 5 and they are: $$A_5, A_6, PSL_2(7), PSL_2(11), PSp_4(3).$$ You should, of course, check that I haven't missed any!