Chapter 3-2 of the Skydiver’s Competition Manual has instructions for establishing state and national records. Appendix B, the claim form, is available for download to the right. A summary:
State records: You’ll need the jump(s) observed by at least one discipline-specific judge (for example, you need a FS judge to evaluate a FS 4-way sequential record attempt) plus one other qualified person listed in Chapter 3 (also listed on the claim form). You only need two authorized signatures to confirm a state record. Not only that, your judge can take part in the jump!
National records: For “performance record” attempts made outside of a USPA National Championships (for example, a formation skydiving large formation record attempt) you’ll need three judges, at least two of them nationally-rated in the discipline involved (one can be a regional judge rated in the discipline). For disciplines documented and graded by air-to-air video, only one judge must actually be on site to validate the performers, document the jump plan and collect the video; the judging itself can be done remotely.
Altitude jumps and most-jumps-in-24 hours claims: You don’t need a judge, although using a judge as an official observer is perfectly acceptable. However, if you’d rather, the pilot’s signature and one other qualified observer (as listed in Chapter 3 and the record claim form) will satisfy the requirement.