Veins are supposed to be in your sclera (the white part of your eyes) but not your cornea (the clear front part). If you have an excessive amount, it's possible that's just normal for you, as different people have varying amounts, just as they have different skin and eye colors, different fingerprints, and so on. If you have more than you used to, it could be because of lack of oxygen or your eyes are dry. You could try using some drops and see if that helps, and if you spend a lot of time on the computer or reading, take regular breaks. I'm pretty sure you'd have to be seriously depriving yourself of sleep for it to affect the vessels in your eyes that much, enough that the stimulants you'd probably have to be using would likely be just as much to blame. In any case, if you think you're not getting enough sleep, then you probably aren't, and you should get more not just for your eyes' sake, but your body's. You can also see if the veins go away after a few days of getting enough sleep, just to rule that out.
On another note, Visine works by making the blood vessels constrict, which means less oxygen and nutrients are getting to your eye. I don't think that makes a big difference, but it's something to consider. Also, due to the mechanism by which it works, the vessels can adapt to it, making it not work as well after a while.