\begin{abstract} {In this paper, we study a class of critical fractional Kirchhoff-type equations involving logarithmic nonlinearity and steep potential well in $\R^N$ as following: \begin{align*} \renewcommand{\arraystretch}{1.25} \begin{array}{ll} \ds \left \{ \begin{array}{ll} \ds \left(a+b\int_{\R^{N}}|(-\Delta)^\frac{s}{2}u|^2\, dx\right)(-\Delta)^s u+\mu V(x)u=\lambda a(x)u\ln|u|+|u|^{2_{s}^{*}-2}u~~~\text{in}~\mathbb{R}^N, \\ u\in H^s(\R^N), \\ \end{array} \right . \end{array} \end{align*} where $a>0$ is a constant, $b$ is a positive parameter, $s\in(0,1)$ and $N>4s,$ $\mu>0$ is a parameter and $V(x)$ satisfies some assumptions that will be specified later. By applying the Nehari manifold method, we obtain that such equation with sign-changing weight potentials admits at least one positive ground state solution and the associated energy is negative. Moreover, we also explore the asymptotic behavior as $b\to 0$ and $\mu\to\infty,$ respectively.}