1. Assuming you are an adult, and no fraud is intended (e.g., changing your name from “Moshe” to “Mosheh” in order to show your support for the offical romanization scheme (which no one uses, but this is for sake of argument) – and you are in the United States (other countries have different and more restrictive rules), then:
2. In many states it is legal to just start using the new name, but you can also get a court order as well. Each state is different. Just using the name may be “bedievad” legal, but a court order will avoid complications when applying for a passport, social security, etc. You probably can get accurate information from the local public library or a local law library.