If you are in an old residential area, might not be a good idea..... too many cesspools, past and present. Tag on the topic though, interested in how to do this.
You can case the hole. Would add more $$, but it would seal the hole. I'd stay away from trying to drill the hole, CLSM it and drill through it.
Have friends that have done it here on Oahu. Gotta find the water table and determine how deep you need to drill. I believe if your close to the ocean, the water is likely to be brackish. They had a company that does well drilling do theirs. They did theirs awhile ago, and last I heard new permitting now makes it pretty hard to drill a well on your own land.
You can drill deeper to get into a different layer of the aquifer.
Oldfart............I can't tell you the specifics of the design and permitting of well drilling, but you can get into some serious cost depending on how deep you need to go, if it needs to be cased, and what you are drilling through.
You can contact Valley Well Drilling. They should be able to give you a rough idea of costs.