Simplest way if you have a rifle to where you can remove the bolt/bolt carrier, etc. Rest it, look through the barrel to center an object as you do your best to treat the chamber, neck and barrel as "ghost rings" without moving anything, look through the scope and adjust to "see what the barrel sees" If possible, use the actual or close to zero distance to do this. A window ledge/corner works well as you can quarter it in the barrel view plus have a level and plumb to work with. With practice you can get very close, I'm usually within .2/.3 at 100. Saves ammunition and time.
Number of rounds? To be efficient, I will shoot one or two, makes adjustments, then 3-5 to confirm, normally recheck on another day. Sometimes on AR's I will add 10 rounds in there, especially with a dot to get a better sample-more of "what I see"
Now, zero check depends on what you are using the rifle for.
.22 short range matches-not a real concern as you get sighters but you should be close anyhow to start.
High Power as an example-you better have a good zero as you only get 2 sighters.
1000 yard match-I had a good zero and tried to shoot less than 5, usually 2-3 for confirmation then went for record as 20 minutes for 20 scored rounds, once you "went for record" no more sighters and some would get the barrel hot as fuck trying to get a zero. My 2-3 were still in, more of "ok, I'm good"
Hunting or work gun-you better know you varying range zero's and impact.
If starting with a fresh build or newly mounted scope, I record the scope settings and once zero is confirmed, reset zero/zero stop and confirm zero again-just I case something didn't go right monkeying with the dials.
As you all know, establish the zero then work those positions to note any changes-and to know your capabilities in that position.
My opinion, haha as every time these threads come up is the math squad comes in to say your sample size needs to be "x". Well true the more the better so to speak but then it gets into stupidity with time and resources spent to where one is mostly frustrated and tired and still haven't come to a conclusion. Once set, 2 consecutive 5 round groups is more than enough, confirm on the next outing and as needed thereafter.
To add-thin barrel hunting guns, keep it to 3 in my opinion, let the barrel cool and confirm. Depending on range space, distance hunting, etc. being able to confirm cb/ccb on numerous days leading up to season is great plus adding in positional with what you have available in the field whether its made made or natural object/props to shoot from.