ag.algebraic geometry - 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
$$ch_{Ycap Z}(mathcal F otimes mathcal G) = ch_Y(mathcal F)ch_Z(mathcal G)$$, 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.