What is the speed of a proton when its kinetic energy is equal to
twice its rest energy?

Here's something to get you started. Clearly we need the relativistic
formula here (the rest energy is a LOT, comparatively; note that if we
use the non-relativistic approximation KE = 1/2 mv^2 we get v = 2c.)

For the kinetic energy to be twice the rest energy, the total energy
must be three times the rest energy. So we need mc^2 = 3(m_0)c^2, or m
= 3 m_0. In other words we want a gamma of 3. You should be able to
work out from this that v/c = sqrt(8)/3, i.e. v = 0.943 c (to 3 s.f.)

Note that it doesn't really matter what sort of particle it is, as the
mass cancels out.