Let $X$ be a CW-complex with skeleta $X^n$. Take $\partial':C_q(X;R)\to C_{q-1}(X;R)$ to be the composite $$C_q(X;R)=H_q(X^q,X^{q-1};R)\xrightarrow{\delta}H_{q-1}(X^{q-1};R)\xrightarrow{i}H_{q-1}(X^{q-1},X^{q-2};R)=C_{q-1}(X;R)$$ where $\delta$ is the connecting homomorphism and $i$ is induced by the inclusion $(X^{q-1},\emptyset)\hookrightarrow(X^{q-1},X^{q-2})$. The quotient $X^{q-1}/X^{q-2}$ is homeomorphic to a one-point union of $(q-1)$-spheres, one for each $(q-1)$-cell of $X$ since the boundary of each $(q-1)$-cell has been collapsed to a point, and a $(q-1)$-cell with its boundary collapsed to a point is a $(q-1)$-sphere. For each $q$-cell $e^q$ consider the attaching map $S^{q-1}\to X^{q-1}$. The composite of this map with the quotient map to $X^{q-1}/X^{q-2}$ defines a map from a $(q-1)$-sphere to a one-point union of $(q-1)$-spheres. Taking the degree of this map in the factor corresponding to a $(q-1)$-cell $e_i^{q-1}$ gives an integer denoted by $[e^q:e_i^{q-1}]$. Now define the differential $\partial'':C_q(X;R)\to C_{q-1}(X;R)$ on a $q$-cell $e^q$ by $\partial''(e^q)=\sum [e^q:e_i^{q-1}]e_i^{q-1}$. How would I prove that $\partial'=\partial''$??
It should be noted that this problem is Exercise 3 in Lecture Notes in Algebraic Topology of which I give a link below for.
The map $H_{n}(X^{n},X^{n-1}) \mapsto H_{n-1}(X^{n-1})$ corresponds to the connecting homomorphism in relative homology, and it is well known/easily shown that for $\mathbb{D}^{n} \mapsto X_{n-1}$, an $n$-cell in $X$ with image $e_{\alpha}^{n}$, then $\delta(e_{\alpha}^{n})$ corresponds to the image of its restriction to $\partial D_{\alpha}^{n} \mapsto \phi_{\alpha}(S^{n-1})$, i.e. you look at where the boundary of the $n$-cell gets mapped to, and we know that the boundary is contained in $X^{n-1}$. ($\phi_{\alpha}$ is the gluing map).
Finally, you can look at the map $H_{n-1}(X^{n-1}) \mapsto H_{n-1}(X^{n-1},X^{n-2})$, which is the quotient map. The $n$-$1$ cycle represented via the image of $\partial D_{\alpha}^{n} \mapsto \phi_{\alpha}(S^{n-1})$ is mapped under the quotient map to $\sum (d_{\alpha \beta}) e_{\beta}^{n-1}$ where $d_{\alpha \beta}$ are the integers you have looking for.