How to show that $f(x,y)$ is continuously differentiable on $\mathbb{R}^2$? I have been given the function $f(x,y)=\begin{cases} \frac{x^3y-xy^3}{x^2+y^2}, \quad \quad (x,y)\neq 0 ; \\ 0  \quad \quad \quad \quad \quad  (x,y)=0.\end{cases}$.
Is it enough to compute all partial derivates and then showing that they are continuous? 

Yes it should be enough to compute all partial derivates and then showing that they are continuous. But notice the following:
Your $f$ is two times partial differentiable, but
$\partial_2\partial_1f(0,0)=-1\neq 1=\partial_1\partial_2f(0,0)$
hence (according to Schwarz's theorem) your partial differentials $\partial_1\partial_2f$ or rather $\partial_2\partial_1 f$ can not be continuous.
