Let $X$ be a smooth scheme (maybe some other condition needed) then the first Chern class map gives an isomorphism between the equivalent classes of line bundle and the linear equivalent classes of divisors: $$ c_1:{\rm Pic}(X)\to {\rm CH^1}(X) $$
Is there an analog of this for higher dimension? Is it true that the Chern class map gives isomorphism between the equivalent classes of vector bundle and the rational equivalent classes of algebraic cycles?
As mentioned in the comments, works well only for smooth varieties. If $X$ is smooth, one has a filtration $\{F^rK_0(X)\}$ on $K^0(X)$ given by sheaves of support of codimension at least $r$. Then, you have the Chern class map $c_r:F^r/F^{r+1}\to CH^r(X)$ and the natural surjective map $\phi_r$ in the other direction. The composite (in both directions) is multiplication by $(r-1)!$ (why you get the isomorphism above for Picard group). This is called Riemann-Roch without denominators. So, if you tensor with $\mathbb{Q}$, $c_r$ is an isomorphism.