Blood is edible to humans. You don't need to adjust a vampire's physiology much to accommodate a diet of blood. If you eat/drink blood your digestive enzymes will break it all down and the useful bits (protein, iron etc) will be absorbed, and the rest will become faeces. The water part of the blood would be absorbed the same way as water in anything else you drink. Your kidneys will remove waste products and excess water from your blood. Your urine would still be urine.
Excessive consumption of blood may lead to iron toxicity. On the other hand, if you have iron deficiency anaemia, you'll get better. Consuming raw blood may put you at risk of illness, as would raw meat. Presumably the vampire's immune system would have adaptations to deal with this. Humans have been cooking food for about a million years now and have adapted somewhat to eating cooked food. Vampires may require some adaptation to deal with the constant high levels of iron without it getting to toxic levels. More of an issue would be the lack of plant foods in their diet, assuming that they're humans/descended from humans and not something that's evolved on an entirely separate branch of the tree of life. But I don't think you'd need to go into that in a story unless you want to write a health-conscious vampire that makes sure they get their five-a-day fruit and veg. I mean, technically they're undead so the lack of fibre and vitamin C probably isn't that much of an issue.
In any case, their pee would still be pee and look like human pee. Unless you want it to look different - maybe the vampire's body is flushing out the excess iron so it looks rusty or something?
