So the Diamond and the Club principles are both combinatorial principles in set theory. They are defined as follows (there are thinner definitions but I stick to this ones is $\omega_1$, as I am sure the ideas in answering this question will suffice to tackle other definitions).
The Diamond principle states that there exists a sequence of sets $\langle A_\alpha\subseteq \alpha : \alpha< \omega_1\rangle$ such that for any $X\subseteq \omega_1$ the set $\{\alpha<\omega_1 : X\cap\alpha=A_\alpha\}$ is stationary in $\omega_1$.
The Club principle is states that there exists $\langle A_\alpha\subseteq \alpha : \alpha< \omega_1\rangle$ such that for each $\alpha$ set $A_\alpha$ is unbounded in $\alpha$ and given uncountable set $X\subseteq \omega_1$ we have that $\{\alpha\in\lim(\omega_1) : A_\alpha\subseteq X\}$ is stationary in $\omega_1$ (this last condition of being stationary can be simplified to being nonempty).
So apparently the club principle clearly follows from the diamond principle, by using the diamond sequence and replacing the sets $A_\alpha$ such that $\sup(A_\alpha)<\alpha$ by something. All I see is that if you enlarge this sets to comply the condition then they might lose their approximation property. I cannot find a solution anywhere online or in books, as all I see is statements that this is clear.
Let $\langle A_\alpha:\alpha<\omega_1\rangle$ witness $\Diamond$. Let $\langle\eta_\xi:\xi<\omega_1\rangle$ be an increasing enumeration of the limit ordinals in $\omega_1$. Let $S=\{\xi<\omega_1:\sup A_{\eta_\xi}=\eta_\xi\}$. For $\xi<\omega_1$ let
$$A_{\eta_\xi}'=\begin{cases} A_{\eta_\xi},&\text{if }\xi\in S\\ \eta_\xi,&\text{otherwise}\;. \end{cases}$$
Now suppose that $X\subseteq\omega_1$ is uncountable. Let $C$ be the set of limit points of $X$ in $\omega_1$; $C$ is a club set, so by $\Diamond$ there is an $\alpha\in C$ such that $A_\alpha=X\cap\alpha$. But $\alpha=\eta_\xi$ for some $\xi<\omega_1$, and $\sup A_\alpha=\alpha$ (since $\alpha\in C$), so $A_{\eta_\xi}'=A_{\eta_\xi}=A_\alpha\subseteq X$. Thus, $\{\xi<\omega_1:A_{\eta_\xi}'\subseteq X\}\ne\varnothing$, and $\langle A_{\eta_\xi}':\xi<\omega_1\rangle$ witnesses $\clubsuit$.