How can I get rid of under eye circles and discoloration around my mouth for even skin tone?

Under eye bags can be permanently corrected with surgery or temporarily camouflaged with fillers.There are great non-surgical options to fill in under eye depressions and hollowness. Using dermal fillers, a liquid eye-lift can be performed. A dermal filler such as Restylane would be placed in the cheek and tear trough to hide the depression or bag underneath your eye.
Surgery would require a fat-repositioning lower lid blepharoplasty. In this surgical procedure, fat from the eye area is moved into the tear trough depression filling the actual depression. 
I hope this helps!

Dark under eye circles can make you look tired and older than you are. I would use a combination protocol that includes topical Melarase creams first, followed by a combination of injectable fillers, fractional laser, and RF to blend out the eyelid hollowness and tighten the skin. 

Recommended Creams: Melarase Brightening Kit

MelaClenz
Melarase AM
Melarase PM
Melapads
Replenish Retinoid Cream
Best, 

Dr. Karamanoukian

Realself100 Surgeon

A combination approach will give the most improvement. Fillers, lasers, and microneedling/PRP can improve under the eyes. Hyperpigmentation can be improved with a combination of lasers, Aerolase, Cosmelan, microneedling/prp, skin lightening, peels, and skin care. Sun protection is essential. I recommend getting a formal evaluation with a cosmetic dermatologist. Best, Dr. Emer