I Have a Green Bluish Dead Vein Under my Left Eye? (photo(

Undereye veins can be removed or ligated.  This is a straightforward procedure performed under local anesthesia.  I would also use Melarase cream for the discoloration of the lower eyes.

The vein needs to be assessed by vein lite and possibly ultrasound. Depending on depth and your skin complexion as well as size of vein, treatment can be planned

Soemtimes NHS docs are reluctant to do procedures unless you go to their private clinics.In the US, such veins under the eyes that are prominent are treated with sclerotherapy, but more likely with microphlebectomy. In a private clinic setting in the UK, expect to pay approximately 800 British pounds. 

It is difficult to see the details of you "vein" from the picture.  I would seek consultation from an Oculoplastics specialist in your area for further guidance.

While this can be treated with laser, some ophthalmologists might consider this a risky procedure given the closeness to your eye.

All veins are under the skin ( basic anatomy).

Get a second opinion for a good diagnosis and plan of treatment.

The picture posted is not very clear. The possibilities are,

Just a prominent vein

or it could be a hemangioma.

Either way it can be treated. some can be treated with lasers and some may require surgery

Really the best treatment for veins in this position if causing concern is microphlebectomy.they must not be injected.the treatment is not available on the nhs.