Nick shops for statistics books for $H$ hours, where $H$ is a random variable, equally likely to be $1,2,3$ or $4$. The number of books $B$ he buys is random and depends on how long he shops according to the joint conditional PMF $$Pr(B=b|H=h) = \frac{c}{h}, \text{ for } b= 1, . . . , h$$ for some constant $c$
Problem: Find a value for $c$
From the definition of joint conditional PMF \begin{align*} P_{B|H}(B=b|H=h) &= \frac{P_{B,H}(B=b,H=h)}{P_H(H=h)} \\ \frac{c}{h} &= \frac{P_{B,H}(B=b,H=h)}{P_H(H=h)} \\ \frac{c}{h} &= \frac{P_{B,H}(B=b,H=h)}{\frac{1}{4}} && \text{since $H$ is equally like to be $1,2,3,4$} \end{align*}
I'm not sure how to calculate $P_{B,H}(B=b,H=h)$ or if my approach is correct.
Just sum the equation you got over $b$ from $1$ to $h$. You get $h\frac c h=\frac {P(H=h)} {1/4}=1$. So $c=1$.