Hi Bailey,

A green discharge from the eye indicated infection and usually goes along with conjunctivitis. You can clean the pus off your dog's eye with a warm washcloth like you have been doing. You can also flush your dog's eye with sterile saline solution like contact lens wearer’s use.   Your vet will have to be consulted in order to get some medicine for your dog. Frequently an antibiotic eye drop is given for this condition. Sometimes Terramycin Ophthalmic Ointment is used for these problems.

You can read more about canine conjunctivitis here.
http://www.petecology.com/Dog_Conditions/Canine+Conjunctivitis.php3

Canine conjunctivitis is a common eye problem. It can happen by itself or indicate an underlying problem. Finding the underlying problem can prevent future bouts with conjunctivitis. Underlying conditions that can cause conjunctivitis is KCS (dry eye, insufficient tear production, upper respiratory diseases such as kennel cough. Usually with respiratory infections there will also be coughing, sneezing, nasal discharge, fever, lethargy and loss of appetite. Conjunctivitis can also be cause by a foreign object in the eye or chronic irritation from a development problem of the eyelid and lashes where the lashes rub constantly against the cornea.

I hope this information is helpful to you.