Let's first prove that $\|\cdot\|$ is indeed a norm.
Notice that if $(P, N)$ is a h.d. of $\mu$, then $(P, N)$ is a h.d. of $c \mu$ for all $c \ge 0$. Also, if $(P, N)$ is a Hahn decomposition of $\mu$, then $(N, P)$ is a h.d. of $c \mu$ for all $c \le 0$. Let $\mu = \mu^+ - \mu^-$. It's easy to see that $(c\mu)^+ = c\mu^+$ and $(c\mu)^- = c\mu^-$ for all $c \ge 0$. Also, $(c\mu)^+ = |c|\mu^-$ and $(c\mu)^- = |c|\mu^+$ for all $c \le 0$. It follows that $\|c \mu\| = |c| \|\mu\|$ for all $c$.
If $\mu \equiv 0$, then $\mu^+ = \mu^- \equiv 0$. If $\|\mu\|=0$, then $\mu^+(X)=\mu^-(X)=0$ and thus $\mu^+ = \mu^- \equiv 0$. Hence $\mu \equiv 0 \iff \|\mu\|=0 \iff \mu^+ = \mu^- \equiv 0$.
Let $(P_\mu, N_\mu), (P_\nu, N_\nu), (P_{\mu+\nu}, N_{\mu+\nu})$ be h.d.'s of $\mu, \nu, \mu+\nu$ respectively. For any $B \in \mathcal B(X)$, we have $\mu (B \cap P_{\mu+\nu}) \le \mu (B \cap P_{\mu+\nu} \cap P_\mu) \le \mu(B\cap P_\mu)$ and $\mu (B \cap N_{\mu+\nu}) \ge \mu (B \cap N_{\mu+\nu} \cap N_{\mu}) \ge \mu(B\cap N_\mu)$. Similarly, $\nu (B \cap P_{\mu+\nu}) \le \nu(B\cap P_\nu)$ and $\nu (B \cap N_{\mu+\nu}) \ge \nu(B\cap N_\nu)$. Thus
\begin{align}
(\mu+\nu)^+(B) &= (\mu+\nu)(B \cap P_{\mu+\nu}) = \mu(B \cap P_{\mu+\nu}) + \nu(B \cap P_{\mu+\nu}) \\
&\le \mu(B\cap P_\mu)+\nu(B\cap P_\nu) = \mu^+(B) +\nu^+(B) \\
-(\mu+\nu)^-(B) &=(\mu+\nu)(B \cap N_{\mu+\nu}) = \mu(B \cap N_{\mu+\nu}) + \nu(B \cap N_{\mu+\nu}) \\
&\ge \mu(B\cap N_\mu)+\nu(B\cap N_\nu) = -\mu^-(B) -\nu^-(B).
\end{align}
It follows that $\|\mu+\nu\| \le \|\mu\|+\|\nu\|$.
Next we prove that $(\mathcal M, \| \cdot \|)$ is complete. Let $(\mu_n)$ be a Cauchy sequence in $\mathcal M$. We have $|\mu_n(B)-\mu_m(B)| = |(\mu_n-\mu_m)(B)| \le \|\mu_n-\mu_m\|$ for all $B \in \mathcal B(X)$. In particular, $(\mu_n (B))_n$ is a Cauchy sequence for all $B \in \mathcal B(X)$. We define $\mu \in \mathcal M$ by $\mu(B) := \lim_n \mu_n(B)$.
Fix $\varepsilon > 0$. There is $N$ such that $\|\mu_n-\mu_m\| < \varepsilon$ for all $m,n \ge N$.
$$
|(\mu_n -\mu)(B)| =\lim_m |(\mu_n -\mu_m)(B)| \le \varepsilon \quad \forall B \in \mathcal B(X), \forall n\ge N.
$$
Notice that
$$
\mu^+(B) = \sup \{ \mu(C) \mid C\in \mathcal B(X), C \subset B\} \quad \text{and} \quad \mu^-(B) =\sup \{-\mu(C) \mid C\in \mathcal B(X), C \subset B\}.
$$
It follows that
$$
(\mu_n-\mu)^+ (X) = \sup \{ (\mu_n-\mu)(B) \mid B\in \mathcal B(X)\} \le \varepsilon \quad \forall n\ge N.
$$
Similarly,
$$
(\mu_n-\mu)^- (X) = \sup \{ -(\mu_n-\mu)(B) \mid B\in \mathcal B(X)\} \le \varepsilon \quad \forall n\ge N.
$$
As a result, $\mu_n \to \mu$ in $\|\cdot\|$. This completes the proof.
We define another norm $\|\cdot\|_\infty$ on $\mathcal M$ by
$$
\|\mu\|_\infty := \sup \{|\mu(B)| \mid B \in \mathcal B(X)\}.
$$
Notice that $-\mu^-(B) \le \mu(B) \le \mu^+(B)$, so $|\mu(B)| \le \|\mu\|$. Then
$$
\|\mu\|_\infty \le \|\mu\|.
$$
Let $(P, N)$ be a h.d. of $\mu$. Then $\|\mu\|_\infty \ge \max \{|\mu(P)|, \mu(N)|\}$. Then
$$
2\|\mu\|_\infty \ge |\mu(P)|+|\mu(N)| = \mu^+(X)+\mu^-(X) = \|\mu\|.
$$
It follows that $\|\cdot\|_\infty$ is equivalent to $\|\cdot\|$.
