In this situation, I would be flying from Richmond Executive-Chesterfield (FCI) to New Kent Co (W96).
Can I land at an uncontrolled airport under class C without talking to ATC?
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It looks to me like New Kent is entirely below the shelf of the class C space. You don't need to go through the Class C Core at all, so I don't see that you need to speak to them. The detour to avoid the core looks minimal
You can see the shelf height shown as 42/14, meaning that the class C space starts at 1400 feet, and goes up to 4200 feet. Just stay below the shelf, at about 1100 feet. At that altitude, pay extra attention to any obstacles like radio antennas! I do see several obstacles going up to 500 / 600 feet!
If you want to let them know of your intentions, I'd just say:
"Potomac Approach: N12345, Departing Richmond Exec, to New Kent, VFR below Class C. On Frequency."
They will likely respond with:
"N12345: Proceed as requested. Maintain VFR"
I would just proceed under the shelf and pick up flight following enroute. If you're under the shelf, you don't have to talk to ATC at all, however I think it would be a good idea as you'll be crossing the extended centerline of all three runways at some point during your flight. Once you get flight following, you can gain altitude and possibly cut straight through the center ring as well.