Natural deduction proof: $A \vee (B \wedge C) ├ (A \vee B) \wedge (A \vee C)$ I assume that I need to set a hypothesis somewhere in the process, but I don't know how. 

1   A V (B Ʌ C)     Premiss
2   B Ʌ C           1 (VE)
3   B               2 (ɅE)
4   C               2 (ɅE)
5   A V B           3 (VI)


The above is the formulas that I have written so far, but I'm clearly stuck and not sure where / how to continue. (I also doubt if line 5 ($A \vee B$) is the right one) 

How should I tackle this problem?
You got the $\lor E$ rule wrong. From $A \lor (B \land C)$ you can not directly infer $A$ and $(B \land C)$ - that would be $\land E$. Instead, you need to open two subproofs where you assume $A$ and $B \land C$ respectively, then when both subproofs yield the same conclusion $(A \lor B) \land (A \lor C)$, this will be the conclusion of $\lor E$, where the assumptions $A$ and $B \land C$ can be discarded. The idea is "If we know that either $A$ or $B \land C$ hold, and from both scenarios proposition $(A \lor B) \land (A \lor C)$ would follow, then we can conclude $(A \lor B) \land (A \lor C)$."  
To continue the proof, you proceed in the same way for $C$ as you did for $B$.


Full proof:
1   A V (B Ʌ C)     Premiss
2   B Ʌ C           1 (VE)
3   B               2 (ɅE)
4   C               2 (ɅE)
5   A V B           3 (VI)
6   A V C           4 (VI)
7   (A V B) Ʌ (A V C)  5,6
8   A               1 (VE)
9   A V B           8 (VI)
10  A V C           8 (VI)
11  (A V B) Ʌ (A V C)  9,10


I'm not correct on the formalism but you were just about there.