What might $$\|u\|_{\mathcal{C}^k(\bar{\Omega})}$$ mean?
$u$ is a sufficiently often differentiable function $\Omega \rightarrow \mathbb{R}$ and $\Omega \subset \mathbb{R}^n$ a bounded domain.
It occurs in the context of difference quotients for finite difference methods:
$$\frac{\partial u}{\partial x_j}(x) = \bigl(\partial^{\pm h}_j u\bigr)(x) + C_1^\pm \quad \text{with }|C_1^\pm|\leq \frac{h}{2}\|u\|_{\mathcal{C}^2(\bar{\Omega})} \text{ if } u\in\mathcal{C}^2(\bar{\Omega})$$ with $$(\partial^{+h}_j u)(x) = \frac{u(x+h\cdot e_j)-u(x)}{h}\\ (\partial^{-h}_j u)(x) = \frac{u(x)-u(x-h\cdot e_j)}{h}$$ so it must be something similar to $$\|u\|_{\mathcal{C}^k(\bar{\Omega})} \overset{?}{=} \max\{\|\partial^{\alpha} u\|_\infty: \alpha \in \mathbb{N}^n, |\alpha|_\infty = k\}.$$
You're almost correct. $$\|u\|_{C^k(\bar\Omega)} = \sum_{|\alpha|\le k} \|\partial^\alpha u\|_\infty$$ Or (equivalent as a norm) $$\|u\|_{C^k(\bar\Omega)} = \sup_{|\alpha|\le k} \|\partial^\alpha u\|_\infty$$ where $\alpha\in\mathbb N_0^n$ is a multiindex and $\partial^\alpha$ is defined as usual.