How fast are the furthest objects we can see moving with respect to our galaxy?
74.3 +/- 2.1 kilometers per second per megaparsec
The most precise measurement ever made of the speed of the universe's expansion is in, thanks to NASA's Spitzer Space Telescope, and it's a doozy. Space itself is pulling apart at the seams, expanding at a rate of 74.3 plus or minus 2.1 kilometers (46.2 plus or minus 1.3 miles) per second per megaparsec (a megaparsec is roughly 3 million light-years). https://www.space.com/17884-universe-expansion-speed-hubble-constant.html (2012)
The FURTHEST object yet discovered was a few years ago. It’s relative rate was not mentioned.
A galaxy 13.4 billion light years away has been spotted by the Hubble Space Telescope.
The new galaxy has been named GN-z11. It takes the crown from EGSY8p7, which set the red shift record at 8.68 — the new galaxy has a red shift of 11.1. That distance means that the light left the galaxy when the universe was in its infancy. The light came just 400 million years after the universe began, 13.8 billion years ago. http://www.independent.co.uk/news/science/most-distant-object-in-the-universe-spotted-by-hubble-space-telescope-shattering-record-for-the-a6911096.html
How it is calculated: http://astro.wku.edu/astr106/Hubble_intro.html