Using an attachment, vacuum the outside of the bed to remove dirt and any particulars that have built up. You can use a lint brush, too, to remove excess hair that has shed and accumulated.
Remove the Cover
Remove the outside cover of the bedding, as this can be washed separately.
Time to Wash
Put both the fabric lining and the cushion into the washing machine. (Make sure the bedding base cushion is machine-washable first.) It's better to wash these two pieces separately from other laundry, but it's also okay to add laundry that is like colored.
Set Your Machine
You'll want to use a large-load setting on your washing machine, with cold water. Always use plenty of pet-safe* detergent to help clean away as much dirt, hair, and grime as possible. For especially dirty beds, wash more than once.
Load Into Your Dryer
Put the fabric lining and cushioning into your dryer along with a pet-safe* dryer sheet. The sheet will remove any final pet hair, and helps to reduce static electricity in the bed. You'll want to set your dryer on low for about 20 minutes. Air-dry the rest of the way to avoid shrinkage.
*Double-check the labels of your detergent and dryer sheets to make sure they're pet-safe.