To expand slightly on  Minhyong's comment, the key facts can be
found in Fulton's Intersection Theory. If you look  at the comment  following
corollary 18.3.2, you'll see an isomorphism (in slightly different notation)
$$ch:K^0(X)\otimes \mathbb{Q}\cong CH(X)\otimes\mathbb{Q}$$
where $X$ is a nonsingular variety, $K^0(X)$ is the Grothendieck group of
vector bundles, $CH(X)$ is the Chow group of cycles mod rational equivalence,
and $ch$ is the Chern character. After mapping this to rational cohomology, you get exactly
the statement you want.

Note that this is false if

 1. you omit the $\mathbb{Q}$, or
 2. if you work on a general (compact) complex manifold because there may not be enough vector bundles or subvarieties. To be clear, I mean that conclusion in cohomology is false:
In Zucker, "Hodge conjecture for cubic fourfolds" Compositio 1977, you can find an example 
of a torus with a nonzero integral $(1,1)$ class which is not a divisor, but it would
necessarily lie in the image of $c_1$.