I'm a little confused about your question. If you just want to know what the summands are there's nothing special about f*ℚX[dim X]. The only way I know of understanding that sheaf is a general algorithm for understanding all semi-simple perverse sheaves. The only fact you use is that a simple perverse sheaf is concentrated in the highest degree allowed by perversity on the largest stratum in its support, and strictly below this degree on all smaller strata (do I need some hypothesis for this? This tends to be the sort of thing I forget).
So, if I have a semi-simple perverse sheaf F, all I have to is look at the restriction of F to each stratum. This will be a complex, whose cohomology in each term is a local system. Perversity includes an upper bound on the degrees that this cohomology can be non-zero. I take the local system in the highest degree allowed by perversity on each stratum, and take the IC sheaves of all those local systems. It happens that in the case of f*ℚX[dim X], this local system has a geometric interpretation (it's the highest degree cohomology of the fiber allowed by semi-smallness), but that's the only thing that's special about this case.