I agree with opting NOT to allow this juice to ferment on its own. I would add bread yeast if you do not have any wine yeast on hand---better outcome.
The amount of sugar to add---well what kind of alcohol content are you hoping to obtain? Remember each cup of sugar added PER GALLON will increase the S.G. by 0.020. SO, let's assume the S.G. of a gallon of tomato juice is 1.003 (completely made that up) and you add 1 cup of sugar to it and it ferments dry, you will have a potential ACV of 3%.
In absence of winemaking experience, you may have a better outcome if you just opt to add vodka or another pure grain alcohol to it and keep it stashed in refrigerator.
Hope that provided some insight.