How do I dye someone's hair when they are bed ridden?

You can also purchase a bottle of "temporary color rinse" sold at beauty supplies or at some pharmacies. Aprox. cost $4.00- They come in all shades, ranging from silver, blonde, brown & blacks. All you have to do is apply on towel dried damp hair with a comb and style as usual. It washes out gradually. Please remember to use gloves when applying to hair as it will stain especially when using a darker shade. Very little is needed and a bottle will last months or up to a year if hair is short. It contains no peroxide- it's a temporary touch up rinse.

Can the person sit up?
I bought my mother a plastic "bonnet" to help keep the water out of her eyes when we wash her hair.
The plastic comes around the head and attaches with velcro strips in the front. This will keep the dye out of eyes. It also has a tube that goes down the back that directs the water away. When you rinse, you could have this empty into a bucket.
If the person cannot sit up, I have seen shampoo devices on medical websites.
I would start with Amazon and Ebay...try "shampoo trays."
Good luck