5. $\frac{1}{192}$.

Since $x^{2}-x+1>0$, we have $k \geqslant \frac{x^{4}(1-x)^{4}}{x^{2}-x+1}$.

Let $t=x^{2}-x+1 \in\left[\frac{3}{4}, 1\right]$.
Then we only need to find the maximum value of $g(t)=\frac{(1-t)^{4}}{t}\left(\frac{3}{4} \leqslant t \leqslant 1\right)$.

This function is clearly a monotonically decreasing function in its domain, hence
$$
k \geqslant g\left(\frac{3}{4}\right)=\frac{1}{192} .
$$
5. If for any $x \in[0,1]$, we have
$$
f(x)=k\left(x^{2}-x+1\right)-x^{4}(1-x)^{4} \geqslant 0 \text {, }
$$

then the minimum value of $k$ is $\qquad$ .