The vicar will probably just invite you to come up to the lectern to read the Psalm. If any words need to be said either directly before or after the reading, the vicar will deal with that. In Protestant churches the reader might say words to the effect of "May God bless the reading of His Word" after the reading is finished. That may not be deemed necessary at a funeral service, though.
I read from Ecclesiastes chapter 3 at my brother's funeral service, held in a very old and beautiful Church of England building. It was an emotional experience and my main concern was to speak clearly and slowly, and to be heard.
Psalm 23 is a beautiful reading and I pray it will comfort you and everyone else there.