I was wondering what is the best tail bound for \begin{equation*} \mathbb{P}\bigg\{\sum_{k=1}^n X_k^4>(1+t)3n\bigg\}\le ? \end{equation*} where $X_k$ are i.i.d. $\mathcal{N}(0,1)$.

What counts as "best"? The smallest tail bound is of course $$ (2\pi)^{n/2} \int_{\{(x_1,\dotsc,x_n) : \sum x_i^4 > (1+t)3n\} } e^{\sum x_i^2 / 2} dx_1 \dotsb dx_n. $$ Presumably you want something simpler. Using standard concentration inequalities, $$ \mathbb{P} \left\{ \X\_4 > \mathbb{E}\X\_4 + s \right\} \le e^{s^2/2} $$ where $\X\_4^4 = \sum_{k=1}^n X_k^4$, and $\mathbb{E}\X\_4 \sim (3n)^{1/4}$. It would take some fiddling to get the best constants, but assuming you're interested in large $t$, you would get an upper bound like $C \exp[c\sqrt{(nt)}]$. 

