I can see my veins running from my upper arm bicep area across my upper chest and onto my other bicep

I would recommend a complete evaluation including ultrasound to determine if there is any systemic or localized problems with your venous return system.  

Bicep to bicep means (I assume) up the inner arm from one side across the upper chest to the inner arm on the opposite side. This could be related to weight lifting and manual labor. A venous Duplex scan can exclude deep issues such as Pagett Schroeder disorder or DVT and determine if there is pathology or if these are normal veins. Hratch L Karamanoukian MD FACSCertified Vein Specialist (ABVLM)

I am very sorry but having a difficult  time visualizing your exact problem. I would recommend showing to a vascular physician. They can determine whether these veins are normal or something you need to be worried about. If there is a vein compressed or blocked it can cause more varices or prominent veins in the areas you describe. However, your description may also be normal. Please show your physician for a clearer answer. 

It is very difficult to comment from your description and limited info.  See a vein specialist for a venous evaluation and to determine if the veins are a normal variant or due to some underlying problem.

That may be "normal" but it may also be related to compression of the vein in the chest.  There is a condition known as Paget Schroetter that can cause this problem.  You may want to see your doctor to determine if a work up for this problem is warranted.