Well, first of all you've lumped in electric grill and driving lights and these are two different things.
So first you need to distinguish which loads you need to supply while the engine is running, and determine if your alternator can handle however many amps that is. Keeping in mind that alternators really should not be run at their full rated load for very long. So if you determined that you needed say 100a continuous, then you'd likely want a 140a or larger rated alternator.
Then, you have to figure out what sort of loads you'll need to supply when the engine is off, and how you will supply them. That's generally done with a deep cycle battery. So let's say the electric grill. A quick search on the net shows me that most are either 1500w or 1750w. Well then 1500w / 12v = 125a. So if you had a battery with a rated capacity of 125 amp*hours, it could run a 1500w grill for 1 hour.
Except it couldn't because you would also need a giant (at least 1500w) inverter and that inverter would itself draw some power. Also, the heavier the load you put on the battery, the lower capacity it has. For example a battery might supply a 10a load for 10 hours, but if you put a 20a load on it, it would last less than 5 hours. Conversely, it would supply a 5a load for more than 20 hours.
So, figure an Optima rated at 65ah with inverter might run a 1500w electric grill for oh...maybe 1/2 hour. More likely it would be less than that. Then it would be completely dead.
The next part is figuring out how you will recharge the battery. You might run your engine for 6 or 8 hours (or more) to get it done, or you might have some solar to help out a little, or you'll probably do what anyone else in that situation does - fire up a generator to run a high capacity battery charger. But if you were gonna do that, then you didn't need to kill your battery or buy a giant inverter.
But yea, you could just get a high output alternator and a giant inverter and use your truck's engine as the generator. Be a bit wasteful of fuel, but it would work.
The most common thing to do is to figure out how much auxiliary battery you can fit, then figure out how much you can recharge it in a day - say by solar or driving a few hours - then cut back on your loads so you don't destroy the (usually expensive) battery by running it dead repeatedly.