I don't know about celebrity but rates are similar to what you might find in hotels. The charge is by item, not the load, and varies by item - that is a dress is more expensive than a pair of shorts. If you want to take the time most ships have laundry machines that would be much more economical.
All I ask is a tall ship and a star to steer her by; John Masefield
Carnival: Glory 2004, Destiny 2008, Splendor 2009, Freedom 2011, Valor 2012, Dream 2013
Celebrity: Summit 2011
Princess: Ruby 2010, 2014, Caribbean 2013, Coral 2014, Regal 2014
Star Clippers: Royal Clipper 2015