I understand that beta particles are radiation and that beta rays are deflected from the electrically charged plates, so they have a charge. But how can I prove that they are electrons?

Unfortunately I don't know how to measure the momentum of a beam of electrons without assuming what you are trying to prove (that is, the value of the electron mass). But you can show that a beam from a cathode-ray tube deflects in the same direction as a beam of radiation from a beta source. The experiment to measure deflection of beta rays is part of the Contemporary Physics Education Project, and it would be simple to set up a similar experiment with a cathode-ray tube.