Yes, there exists such a treatment by Deligne, see "Cohomologie a supports propres", SGA4, Tome 3, Lect. Notes Math. 305, subsections 1.2.1-1.2.2.  Basically, what one needs is that that for any object X in D there exists a morphism X→Y in D with a cone in C such that for any morphism Y→Z in D with a cone in C there exists a morphism Z→W in D with a cone in C such that F(Y)→F(W) is an isomorphism.  Then one defines RF(X) as F(Y).