Let $K$ be a compact Hausdorff and $C(K)$ be a Banach algebra of continuous function on $K$ such that $\textbf{1} \in C(K)$ and such that $C(K)$ separates the point of $K$. I am trying to show that $\delta : K \rightarrow \Phi_{C(K)}$ is a homeomorphism where $\Phi_{C(K)}$ is the space of characters on $C(K)$ and $\delta(K) = \delta_k$ the evaluation at $k$.
So far I showed that $\delta$ is a bijection between $K$ which is compact and $\Phi_{C(K)}$ which is Hausdorff.
All I need to do is to show that $\delta$ is continuous but I don't really know where to start.
That's the easy part. The topology you need to consider on $\Phi_{C(K)}$ is the weak* topology; that is, pointwise convergence.
So, if $x_j\to x$ in $K$, you want to show that $\delta_{x_j}\to\delta_x$. This means that $\delta_{x_j}(f)\to\delta_x(f )$ for all $f\in C(K)$. But this is $f(x_j)\to f(x)$, which is precisely the continuity of $f$.
Conversely, if $\delta_{x_j}\to\delta_x$ you have $f (x_j)\to f (x) $ for every $f\in C (K) $. Then $x_j\to x $ by Urysohn's Lemma.