Consider the graph as given in the third page (pg.99) in the link below.
https://www.math.cornell.edu/~hatcher/AT/ATch2.pdf
We define a homomorphism $\partial: C_1 \to C_0$ by sending each basis element $a,b,c,d$ to $y-x$, the vertex at the head of the edges minus the vertex at the tail. Thus we have $$ \partial(ka + lb + mc + nd) = (k+l +m +n) y - (k+l + m +n )x$$, and the cycles are precisely the kernel of $\partial$.
Apparently it is a simple calculation to verify that $a-b$, $b-c$, and $c-d$ form a basis for this kernel.
Unfortunately, I do not know what this calculation would be. It would be nice to see an example of a such a calculation in this easy example.