The short version is, most people don’t bother RPing much about their arrows. Here’s the long version:
A typical RP with an archer would be something like: Assess your bow, string and arrows to make sure they’re still sound prior to battle RP starting (bow/arrow failure is a thing and it can be nasty if it happens). When you’re RPing an attack, make sure the field in front of you isn’t obstructed by a friendly melee fighter. If not, you pull an arrow with your index finger and thumb at the fletching end of the arrow. Then you nock, aim and loose. If a friendly is standing in my way, I’ll RP taking a step to the side. You can RP shooting stationary or while moving. I like to position high when I can.
After the RP, if you want to determine how many of your arrows are still good, you can do a roll. If you shot 6 arrows, then roll 1d6 to see how many are found or still in good condition.
If you want to go the magical route, there are special trick arrows that Hawkeye or Green Arrow use. These can be fire, freeze, explosive, smoke or blinding flash, sonic, net, zip line or grappling hook, tranquilizer, electric shock, listening devices, flare, magnetic, blunt tip, etc. If I RP any of these, I like to run it by the DM for approval and I don’t like to use more than a couple different specials. I feel like engineering or enchanting professions would be involved in creating these and that they’re 1-use only and expensive.
Details:
Bows come in a few different types in-game. There’s long bow, recurve and crossbow (I don’t remember many compound bows, but I think we have at least a couple).
For long bow, recurve and compound bows, you’ll be using arrows in a quiver that’s either on your back or hanging from your belt. The assumption is you’ll have around 12-20 arrows in that quiver at any given time (I usually don’t count them ICly, but I also don’t RP attacking more than 12x in a battle). If you’re a dedicated archer, those arrows are either hand-made by you or by a fletcher who is familiar enough with you to measure your draw length (how far you pull back the bow string) to know how long to make your arrows. The arrows will be specific to you in this case, and you may have your own markings/coloring on the arrows to be able to find and identify them after a battle.
The arrows will have (typically) 2-3 feather bits called fletchings. The notched end of the arrow is set against the string, then you nock the arrow - pull back the string and arrow to an anchoring point (a lot of people use the corner of their mouth as their string pull length), aim and let loose.
If you have time to set up, you might take the arrows out of your quiver and sink them tip first in the ground in front of you - easier to grab quickly for a faster nock but not necessarily good for the arrow. If you’re aiming at something far away, you aim your direction first, then angle the bow upward.
If you’re not using a crossbow or a compound bow, you won’t stand for long periods of time holding your bow as nocked. If you do, your arm will get tired. If you want to get fancy, you can nock more than one at a time and loose them (you hold them close together when you do and I roll for each arrow). There’s also string walking, which involves moving your hand lower than where on the string the arrow is nocked (if that makes sense) which changes where you’re aiming.
A crossbow uses bolts, not arrows. And your nocking is different. You’ll either face the crossbow down toward the ground and step into it to pull the string back, or some have cranks to pull the string into place. Then it stays nocked until you shoot it. Nock, load your bolt, aim and fire.
Other random bits: A crossbow is more accurate than a recurve because it doesn’t drop during flight as much. A compound bow has pulleys on top and bottom that allow you to hold your bow as nocked for longer periods of time. An archer may make small markings on their bow that show distance measurement angles for accuracy. They may also ask someone to spot for them and tell them if the shot went short, too far, etc. Winds can blow your arrow off course.