how to make the category of chain complexes into an $\infty$-category

As everybody's said, there's an obvious thing to do. As Yosemite Sam cites, it's done in Section 13 of the ArXiv version of DAG I -- you think of chain complexes as enriched over simplicial sets via Dold-Kan, and then apply the nerve construction.

But there's an explicit thing you can do for any dg category, and I find it useful because it's given in terms of formulas. Moreover there's an obvious (if tedious) way to generalize this formula for any $A_\infty$-category so it's a cool thing to know. It's in the latest (February 2012) version of Higher Algebra. Since chain complexes obviously form a dg-category, this explicit method might be what you're looking for in case you want to produce some simplices in your quasi-category.

Specifically, Construction 1.3.1.6 tells you how to get a quasi-category from any dg category. Then Construction 1.3.13 and Remark 1.3.1.12 should convince you that it's equivalent to the "Dold-Kan + Simplicial nerve" construction cited by everybody else. (Lurie summarizes this equivalence in Proposition 1.3.1.17.) I would write out the formulas here but I don't want to re-TeX the long discussions. So here's at least a link to the latest Higher Algebra.

For any simplicial model category $C$, let $C^0$ denote the fullsubcategory on its fibrtant and cofibrant objects. It may be considered as a simplicial category via its simplicial-enrichment. Via the corner axioms for this enrichment, it follows that the Hom-complexes of $C^0$ are Kan complexes, so that $C^0$ is a fibrant simplicial category in the Bergner model structure on simplicial categories. Then apply the homotopy coherent nerve to $C^0$ as you suggest. Since it is the right-Quillen pair of the Quillen equivalence between the Bergner model structure on simplicial categories and the Joyal model structure on simplicial sets, the result, $N_{hc}\left(C^0\right)$ will be a fibrant simplicial set in the Joyal structure, i.e. a quasi-category.

[ 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!