I'd take the view that a real-valued Radon measure "really is" a continuous linear functional $\Lambda:C_{00}(X,\mathbb R) \rightarrow \mathbb R$. Here $X$ is a locally compact space, and $C_{00}(X,\mathbb R)$ is the collection of compactly supported continuous real valued functions on $X$. By "continuous", we mean that for each compact $K\subseteq X$, there is a constant $c>0$ such that
$$ |\Lambda(f)| \leq c \|f\|_\infty $$
for all continuous $f$ supported in $K$. (For example, with $\Lambda$ given by integrating against $\sin(x) \ dx$, we can take $c$ to be the Lebesgue measure of $K$).

So you could just work abstractly with such linear functionals. (This is going to be a sketch, but if you've done a course on abstract integration, nothing should be too great a surprise). Then you do the usual trick: define
$$ \Lambda_+(f) = \sup\{ \Lambda(g) : 0\leq g\leq f \} $$
for all $f\in C_{00}(X,\mathbb R)$ which are (pointwise) positive. The supremum is positive (take $g=0$) and finite because $\Lambda$ is continuous.
Notice that $\Lambda_+$ is positive homogeneous, and clearly $\Lambda_+(f_1) + \Lambda_+(f_2) \leq \Lambda_+(f_1+f_2)$. Conversely, if $0 \leq g \leq f_1+f_2$, then let $g_1=\min(g,f_1)$ and $g_2=g-g_1$, so $0\leq g_1\leq f_1$, $0\leq g_2\leq g_2$, and $\Lambda(g) = \Lambda(g_1)+\Lambda(g_2)$. It follows that $\Lambda_+$ is additive.

Then extend $\Lambda_+$ to all of $C_{00}(X,\mathbb R)$ by $\Lambda_+(f) = \Lambda_+(f_+) - \Lambda_+(f_-)$. That $\Lambda_+$ is linear will follow from *exactly* the same proof which shows that the Lebesgue integral is linear (having first defined it for positive functions) from pretty much any course on integration.

Then define $\Lambda_-(f) = \Lambda_+(f) - \Lambda(f)$, so
$$ \Lambda_-(f) = \sup\{ \Lambda(h) : -f\leq h\leq 0 \}\geq 0 $$
(set $h=g-f$). So $\Lambda_+$ and $\Lambda_-$ are *positive* linear functionals on $C_{00}(X,\mathbb R)$ and are hence given by positive Radon measures.

I don't know a self-contained reference-- as I say, I was exposed to these ideas in courses on integration theory, and basic functional analysis. I like Rudin's book "Real and complex analysis" for the basics on positive Radon measures.