Similar to How one shows $K_{n,m}$ is not planar. and Prove that $K_{3,3}$ is not planar.
How does one show that $K_5$ is not planar?
One could try demonstrating non-planar drawings, but how do you know that you've not missed a planar drawing.
One could also calculate $V=5$ and $E=10$, but what is $F$?
As Lorenzo said, take the incidence matrix $M=(m_{e,f})$ of a planar connected graph with $n\geq 3$ nodes. Each edge $e$ is surrounded by $\leq 2$ faces and each face $f$ is surrounded by $\geq 3$ edges. So each row of $M$ has most 2 1's and each column has at least 3 1's. Thus $3|F|\leq 2|E|$. By the Eulerian polyhedra formula, $|V|-|E|+|F|=2$, it follows that $|E|\leq 3|V|-6$. So each planar connected graph with 5 vertices has at most 9 edges. But the $K_5$ has ${5\choose 2} = 10$ edges.