It's well known that if E is a vector bundle with Chern roots $a\_1,\ldots, a\_r$,
then the Chern roots of the $p$th exterior power of E consist of all sums of $k$ distinct $a\_i$'s. I would like to say the same is true if E is just a torsion-free coherent sheaf on $P^n$. It seems non-obvious, though, maybe because an exterior power isn't generally an additive functor. 


Presumably this is either false or also well known, but I can't find a reference.



A reference: the splitting principle for exterior powers of coherent sheaves?
My guess would be that the formula you want does not extend to the case of coherent sheaves. As indicated in Mariano and David answers (which has unfortunately been deleted), the best hope to compute is via a resolution $\mathcal F$ of $E$ by vector bundles. In general, for 2 perfect complexes $\mathcal F, \mathcal G$ of vector bundles, there is a formula for the localized chern classes 
$$ch\_{Y\cap Z}(\mathcal F \otimes \mathcal G) = ch\_Y(\mathcal F)ch\_Z(\mathcal G)$$, with $Y,Z$ being the respective support. Unfortunately, this only gives the right formula for the "derived tensor product".


So to mess up the formula, one can pick $E$ such that $Tor^i(E,E)$ are non-trivial. I think an ideal sheaf of codimension at least 2 would be your best bet for computation purpose.


