Little Bumps Under My Eyes

How old are you?

They may be tiny cysts, a rash, or an irritation.

Do you put eye cream around your eyes at night?

It'd be next to impossible to say for certain without seeing them. Perhaps you should ask a medical professional? That's what I'd do, if it were me.

The dark circles are sometimes something that can indicate nutritional issues like and allergy or vitamin deficiency. For others with this problem (I've read this) it can mean that there's tiny blood vessel or capillary damage due to a very thin layer of skin.

If you're concerned, I'd ask a doctor (perhaps a dermatologist) next visit.