For compact orientable manifolds, this is accomplished by the notion of a [spectral triple](https://ncatlab.org/nlab/show/spectral%20triple).

See the question https://mathoverflow.net/questions/191720/commutative-spectral-triples for additional information, including a [precise statement](https://mathoverflow.net/questions/191720/commutative-spectral-triples/191743#191743) of the theorem.

See also the answer to https://mathoverflow.net/questions/16833/noncommutative-smooth-manifolds/117813#117813 for additional remarks about orientability.

As a side remark, if one is willing to use smooth functions instead
of continuous functions, then there is a very satisfactory answer:
the resulting contravariant functor from second countable Hausdorff smooth manifolds to commutative real algebras is a fully faithful functor, i.e., manifolds are (contravariantly) identified with a full subcategory of real algebras.  This is known as [Milnor's exercise](https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras).
See also the answer https://mathoverflow.net/questions/14877/how-much-of-differential-geometry-can-be-developed-entirely-without-atlases/14890#14890 for a discussion on how to characterize the essential image of the embedding functor.