[ Edit: Section 13 of DAG I had everything I was looking for http://arxiv.org/pdf/math/0608228v5.pdf ]
I think I now partially understand why I'm confused (I'd like to thank David's answer for providing a more high-brow perspective, I'm sure it will be useful to me as soon as I try and understand stable/derived categories)
The $\infty$-category one should obtain has for vertices complexes $A,B, \ldots$, the 1-simplices are given by chain maps $A \to B$, the 2-simplices are given by maps $A \to B \to C, A \to C$ (not necessarily commuting) together with a homotopy, the 3-simplices are given by maps $A \to B \to C \to D, A \to C, B \to D, A \to D$ together with a homotopies and homotopies among homotopies and so on (perhaps I got something wrong but you get the idea).
I'm pretty sure this is the coherent nerve construction for simplicial categories but I need to understand it better first.
This should give the right thing, an $\infty$-enhancement of the category of complexes such that $\pi_0$ of it is the homotopy category of complexes (so no resolutions and no model categories were harmed in the process).
If someone corrects and/or has a better way of writing this please do so!