I would like to find a sequence $(u_{n})$, where $u_{n} \in L^{p}(\mathbb{R}^{N})$ for all $p\geq 2$ and such that $u_{n}(x) \rightarrow 0$ a.e in $\mathbb{R}^{N}$ and $|u_{n}|_{L^{2}} \rightarrow +\infty$, but $|u_{n}|_{p} \leq C$ for all $n\in \mathbb{N}$ and for all $p>2$.
In general, can we derive some property of $(u_{n})$ in $L^{2}$ through the properties of $(u_{n})$ in $L^{p}$ for $p> 2$?
The following function verifies your assumptions $$ u_n(x) =\frac{1}{(\ln(n+1))^{1/4}} \frac{\mathbf 1_{|x|<n}}{(1+|x|)^{N/2}}. $$ Indeed, the fact that $\|u_n\|_{L^p} < C$ and $u_n\to 0$ a.e. are not difficult to see (even more, $u_n\to 0$ in $L^p$ for $p>2$ actually).
Moreover, $$ \|u_n\|_{L^2}^2 = \frac{C_N}{(\ln(n+1))^{1/2}} \int_0^n \frac{r^{N-1}}{(1+r)^{N}}\,\mathrm d r \underset{n\to\infty}∼ C_N\ln(n)^{1/2} $$
In general indeed, it is not possible to get properties of $u_n$ in $L^2$ from properties in $L^p$, except for instance on bounded domains.