Completeness of a space is a metric notion (in the most usual definitions). In the case at hand means that each Cauchy sequence of Hilbert space vectors converges in the topology inherited by the inner product structure (that also defines naturally a metric).
The other fact, i.e. that each vector can be written as a linear combination of a particular set of vectors, is a quite different notion and depends really on the precise definition and purpose. It is not called completeness however, but usually "existence of a (some adjective) basis". For Hilbert spaces, it is the "existence of an orthonormal basis" and the result is that each vector can be written as an infinite linear combination of elements of the basis, and the sum converges in norm.
the first concept requires only the notion of a metric (i.e. distance between objects);
the second concept relies on the vector space structure and also on the inner product structure of the space (to define infinite linear combinations and orthonormality).