I used to train at a Class C and this brought back some memories! Here is how we conducted the runups with ground and hopefully it will be helpful.
All of the GA airplanes parked in one large area. I would normally call ground once I started up and would request a taxi clearance and also state I needed a runup.
Ground, Cessna 12345 with alpha, ready to taxi from GA ramp, runup required
There were two runup pads on the field. One was at the end of runway 28, so if going to that end ground would simply clear me to taxi to runway 28 and do my runup at the end of runway
Cessna 12345, Ground, taxi to runway 28 via A and do your runup at the end of A.
After doing the runup here, I pulled up to the hold short line, switched to tower, and advised ready for departure.
The other runup pad was near the end of runway 10 but was off to the side and butted up to some hangars. If using this pad, ground would ask for me to call before continuing taxi. I think they asked this because I had to merge back onto the original taxiway in order to get to the runway, and they did not want to risk me screwing up their traffic flow.
Cessna 12345, Ground, taxi to runway 10 via A and do your runup at the hangar pad. Advise when you are finished with the runup.
Once finished, I would advise ground and they cleared me to taxi the rest of the way. Again, once I arrived at the hold short point I switched to tower and advised ready to depart.
...The moral of the story is if ground simply gave me a taxi clearance to the runway and no specific instructions to call them, I would complete the runup and taxi to the hold short line without further calls and then switch to tower. I only made calls to ground along the way if they asked me to.