Hawaii 5.0 has pointed out the most important tip which is the weave ahead. I also think you should weave using one hand below and one hand above the strings so you can more efficiently press the cross through.
Let me add that watching TV while you are stringing is bad practice. IMO, you are better off paying attention to what you are doing, which if you practice this in general, will also help your tennis game.
Finally, if you are using a drop weight like a Klipper, you shouldn't get anxious about having the bar exactly horizontal. As many have pointed out, even a 5% error only results in a fraction of a lb difference in tension. Add to this the fact that after a few minutes of hitting the tension is pretty well evened out over the entire string bed.