Harry, one doesn't 'register to vote'. Once a person has Mexican citizenship, the person needs to go to the nearest INE office (there's one in Jocotepec) to apply for a credencial--a national photo identification card that is the ID needed in order to vote and to identify oneself as a Mexican citizen. Mexican citizens are asked to show that ID many more times than one might think and it's important to carry it with us at all times. Usually one goes to the INE to apply for it and then returns approximately 2 weeks later to pick it up. In order to apply, one needs to take the Acta de Naturalización and a comprobante de domicilio to the INE.
There's no charge for the credencial, and no party affiliation is involved.