How to Get Rid of Blue Chest Veins?

It is the rare patient who has spider veins on the chest so lasers are not likely to hel with these veins. We successfully use foam sclerotherapy for these veins which are almost always varicose veins (blue, under thin white skin). We don't inject breast tissue. I recommend that you see someone who does a lot of foam sclerotherapy for their expertise.

If the vessels are small and superficial (more reddish purple) than the pulsed dye laser can be used to reduce the appearance of vessels on the chest.  If the vessels are deeper and more blue, then a longer wavelength laser, such as the long pulsed Nd: YAG laser, can be used to reduce their appearance.  An evaluation by a dermatologist or plastic surgeon with experience in lasers is indicated.  Sclerotherapy, which involves injecting an agent into these small vessels to cause them to close off, may also be an option.

Facial veins respond to Laser better than any other method or treatment.
Facial veins respond better than leg veins.
Research the lasers and the person using the laser. ONLY MEDICAL DOCTORS SHOULD USE LASERS

The major modalities for treatment of unsightly veins are sclerotherapy and lasers. Sclerotherapy is generally reserved for treatment of leg veins, while lasers are most often the first line therapy for veins on the trunk, neck or face.
The most effective and most popular lasers for vein treatment are the Pulsed Dye Laser (595nm), long pulsed Nd:Yag lasers (1064nm), and the KTP (532nm) laser. I generally use the Pulsed-Dye or the KTP for red to red-purple, superficial veins. The Nd:Yag lasers are usually best for blue-purple, deeper veins.
Of course, being properly evaluated by a reputable laser surgeon, generally one who is Board Certified in Dermatology or Plastic Surgery, to evaluate your individual case is most important to determine the therapy best for you.