I had assumed for a long time that all professional locksmiths know how to drill the shear line of a pin tumbler lock, but when I had to replace a very expensive commercial duty Schlage lockset after an employee, who'd been hired as an experienced locksmith, drilled incorrectly and wiped out the mechanism behind the plug entirely.
If it is not possible to pick a pin tumbler lock, and no other entry seems open to you, drilling the plug is the accepted "last resort" method of entry. If it's done correctly, there is no damage to the lock mechanism just to the plug, which can be replaced easily and inexpensively.

My employee had the impression somehow that drilling is to be done at the center of the keyway, and when a half hour of drilling and cranking on the knob failed to open the door, he just kept drilling deeper and deeper. Annihilation of the lockset ensued.

The shear line is where you want to attack the plug. This keeps the drill bit within the plug itself, but high enough toward the shear line that the pin tumblers will be eliminated by the bit and eventually nothing will be left to prevent the plug from rotating in the shell.

Begin by placing a center punch at dead center vertically but above the keyway horizontally, as shown below.

Make a good dent with the center punch to guide the drill bit. Use a bit no larger than 1/8" and choose a brand new one if possible. Brass is relatively soft, but the pins will rotate when struck with the spinning bit and they will quickly dull a used bit to the point of uselessness. A new one has a much better chance of doing a clean job.
Keep the drill level. Continuously monitor yourself or have a partner watch as you travel inward with the drill. If you drill at an angle, you'll make the job harder and you could stray beyond the limits of the plug and do damage to the lock, rendering it incapable of accepting a new plug.

As the bit passes through the initial metal stock of the lock face, you'll feel a moment of less resistance. It will then encounter the first pin tumbler (more resistance). Put heavy pressure on the drill motor as each successive pin tumbler is encountered. Count them as the bit first encounters them and then chews through them. Most locks have five tumblers some have six. After counting the destruction of five tumblers, stop. Clean out the hole as best you can, insert a heavy screwdriver with a blade that fits well in the keyway, and turn the plug. It may require alternately forcing the plug one way and then the other to coax debris out of the shear line. If this doesn't work after a while, drill the hole again, going no further than before, but slightly reaming the hole larger. Try to turn the plug again.

If the plug will not turn, you may have a 6th tumbler to get rid of. Start drilling again, applying force carefully. If the bit does not almost immediately penetrate another tumbler, STOP. You're probably drilling through the back of the plug now and you don't want to risk going any deeper.

If this is the case, it's simply going to require more persistence in cleaning the brass debris out of the drilled hole. The plug WILL turn if you have placed the drill point in the right place and have drilled out all the tumblers.