[Disclaimer: I am not a physicist, so what follows may be total garbage.]
The total energy (rest energy plus kinetic energy) of a mass m moving at speed v is \(\displaystyle E = \frac{mc^2}{\sqrt{1-v^2/c^2}}\). You want this to be twice the rest energy. So \(\displaystyle \frac{mc^2}{\sqrt{1-v^2/c^2}} = 2mc^2\), which gives \(\displaystyle v=(\sqrt3/2)c\).
