You can try the paper by Dolgachev Weighted Projective Varieties

In Section *3.3*, page 54 there is the treatment of the dualizing sheaf of a weighted projective space or, more generally, of a quasi-smooth weighted projective intersection (see in particular Theorem 3.3.4, page 56)

Regarding Riemann-Roch, one needs to make some corrections to the usual formula, because of the presence of singularities. The exact expression of these corrections, in the case where the singularities are all isolated, can be found in M. Reid's paper *"Young Person's Guide o Canonical Singularities"*, Algebraic
Geometry, Bowdoin 1985, ed. S. Bloch, Proc. of Symposia in Pure Math.
46, A.M.S. (1987), vol. 1, 345--414. See in particular Chapter III *"Contributions of $\mathbf{Q}$-divisors to $\mathbf{RR}$"*.

Unfortunately I could not find Reid's paper online. However, I'm quite sure that some electronic copies are circulating.