The basic requirements are:
- Private pilots must have at least 500 hours of total time.
- PIC is required to comply with the requirements of Part 136 Subpart A, including providing a full safety briefing, and where appropriate, carry overwater gear.
- Pilots are limited to 4 such events for charitable/non-profit causes per calendar year.
- And the flights themselves are limited to day VFR conducted within a 25 mile radius of the departure airport, and the departure and arrival airports must be the same.
- No aerobatics or formation flight.
- You are also supposed to notify the local FSDO at least 7 days prior to the event.
You can find a nice summary of the regulatory requirements here:
Often times, pilots offer a single ride to a charity auction. (As opposed to participate in a day(s) long fund raising event.) In that case, it is not clear what the reporting requirements to the FSDO are. But since it is just paper work, I think you would be safe if you were to report the proposed date (and any alternates) of the ride itself.
Also, since the general public does not always understand what is involved, how safe flying is, and how much fun it can be, it might be a good idea to put up a webpage describing the pilot, the aircraft, the proposed route/activities, etc. Philip Greenspun has a nice example here.
Finally, often charities will ask you about what the actual "value" of the ride would be. While that might be simple as taking the operating cost of your aircraft for the time involved, it might be more accurate to provide a market value where possible. For example, when I offered a 45 minute aerial sightseeing tour of the Hudson River around Manhattan, I noted that commercial operators charge upwards of $200 per person for 20 minutes.
When I've done this, I've been pleasantly surprised by what people are willing to pay to go on an airplane ride (understanding that part of that is motivated by their desire to help the charity), and many charities think that such rides are a cool item to offer as part of their fund raising events.