Ionization happens when an electron in an atom gains enough energy to escape, where "enough" here means "greater than or equal to the ionization energy of the atom". This can occur in many ways. An atom in isolation can be ionized by very strong electric fields, when the electron's wavefunction distorts enough to allow for tunneling outside the atom. Ionization can also be achieved by hitting the electron with something of sufficient energy, such as a photon or another electron. In either case, what you need is for energy to be transferred to the atom. Simply applying a potential difference without regard for electric field strength will not suffice.
If I'm correct in assuming your second question is actually, "How do you find the kinetic energy of the final products of ionization?", then the answer depends on particular method of ionization. In the case of ionization by radiation, if we assume that the atom remains pretty much stationary in the process,* then the kinetic energy of the ionized electron is simply the difference between the photon energy and the ionization energy. In the case of ionization via collision, treating the collision as a semiclassical momentum transfer between an electron in an atom and a colliding particle should yield reasonable results. In the case of electric field ionization, the electron's kinetic energy as a function of position can be calculated numerically by assembling the proper time-independent Hamiltonian. For large ionized ensembles, the temperature is the ensemble average of the kinetic energies of all species present, just like in normal thermodynamics. One must, however, be careful which temperature one needs, as, for example, the temperature of electrons and of ions in a typical plasma are very different.
*The atom is so much heavier than the electron that this is typically a good assumption. The same assumption is made later, in the electric-field-ionization case.