In this paper, we study the multiplicity of solutions to a class of Kirchhoff-type equation with critical growth $$ -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2dx\right)\Delta u+V(x)u=\lambda h(x)f(u)+g(x)u^5 \quad \mbox{in }\mathbb{R}^3, $$ where $a$, $b>0$, $\lambda$ is a positive parameter and $f$ is a continuous nonlinearity with subcritical growth. Under suitable conditions on the potentials $V(x)$, $h(x)$ and $g(x)$, we prove the multiplicity results and investigate the relation between the number of solutions with the topology of the set where $g$ attains its maximum value for small values of the parameter $\lambda$. The proofs are based on Nehari manifold and Lusternik–Schnirelmann theory.