How do finite fields work? I know that a field must satisfy a set of axioms, the one that is causing me most discomfort is closure under addition. All the roots of $X^q-X$ are all the elements of a finite field of order $q$. However, I tested this with $X^5-X$, its roots are ${1,-1,0,i,-i}$ however, if I add $i+i=2i$ and $2i$ is not in the field, failing the axiom of closure, how do finite fields work then? 

You're not taking the roots in the complex numbers, you're taking the roots in the finite field itself. In the finite field $\mathbb{F}_5$ (the integers $\bmod 5$), there is in fact an element that deserves to be called $i$ (in the sense that it squares to $-1$), namely $2$, and $2 + 2 = 4$ is also in the field (if you like, we have $i + i = -1$; sounds weird, but it all works out $\bmod 5$). 
