Rather than be given a proof, I'd like to know why my proof is wrong. (It's clearly wrong because I got the wrong final answer.)
We build an injective resolution of $\mathbb{Z}$; one possibility is
$$ 0\rightarrow\mathbb{Z}\rightarrow\mathbb{Q}\rightarrow\mathbb{Q}/\mathbb{Z}\rightarrow0\dotsb. $$
Indeed, this is exact, and $\mathbb{Q}$, $\mathbb{Q}/\mathbb{Z}$, and $0$ are injective $\mathbb{Z}$-modules because divisible. (We consider $\mathbb{Q}$, $\mathbb{Q}/\mathbb{Z}$, and $0$ as constant sheaves on $S^1$.)
The corresponding complex is
$$ \dotsb\rightarrow0\rightarrow \mathbb{Q}\overset{d^0}{\rightarrow} \mathbb{Q}/\mathbb{Z}\overset{d^1}{\rightarrow} 0\rightarrow \dotsb. $$
Applying the global section functor $\Gamma(S^1,\cdot)$, we get
$$ \dotsb\rightarrow\Gamma(S^1,0)\rightarrow \Gamma(S^1,\mathbb{Q})\overset{\Gamma(S^1,d^0)}{\rightarrow} \Gamma(S^1,\mathbb{Q}/\mathbb{Z})\overset{\Gamma(S^1,d^1)}{\rightarrow} \Gamma(S^1,0)\rightarrow \dotsb, $$
which just becomes our original sequence since we're dealing with constant sheaves.
Therefore, $H^1(S^1,\mathbb{Z})=\frac{\ker d^1}{im d^0}=\frac{\mathbb{Q}/\mathbb{Z}}{\mathbb{Q}/\mathbb{Z}}=0$.