Greasy hair & skin can mean seborrhea:
Skin Disease (Canine Seborrhea) in Dogs | petMD
Seborrhea is a secondary condition & can often led to skin infections. Princess' is triggered by allergies & she's allergic to Lamb also, it makes her itching & seborrhea worse. She has a mixture of both, greasy hair & dry scabby looking flakes. She's also had numerous skin scrapings to rule out other skin issues.
I used to use RX shampoo & conditioner vet prescribed, but it didn't do much. I tried Hylyt shampoo & conditioner, it has fatty acids & is coconut cleanser based...wonderful & not RX. I bathe her weekly to keep her skin & hair clean & wash her paws daily to remove any allergens. Lately I've been using baby wipes & that seems to work well too.