I'm so sorry to hear of her passing. I'm sure she saw a strength in you that you were not aware you have for her to select you to read Psalm 23. While I've not been in your position, I would like to think that if I had some tissues at hand, I would tell the audience something lighthearted about your friend and with a smile on your face begin the reading. If you need a break during the reading, take your break and no one will complain. If you feel you cannot complete the reading, have a plan 'B' in place for another person to take over if they can. My guess is you'll be able to complete the reading without needing any help at all.
Its like the folks who sing at funerals. I guess they put themselves into a place where they know they have to 'perform' and they belt out the song without a tremor and its usually one of the best performances of their lives! I'm not saying that you have to 'perform', but if you try to put your mind to the fact that this is a proud moment that your friend entrusted you with, you'll do just fine.
Oh, and take a deep breath before you begin and make sure to let it out!
Perhaps if you can speak with a Clergy person, they can also advise you on getting through this very emotional time. Later, after you have healed from this sad time, I'm sure you'll look upon this moment as a very nice but somber, yet proud experience. Hopefully one, that later you can smile about and remember your friend in a very nice way.
Take care,
Michelle