The average adult contains about 5 liters of blood. The average drunk contains lots of alcohol. So could you, in theory, have a cheap—if messy—night out by turning vampiric for the evening?
Fortunately, Randall Munroe has contemplated this question at length. He first works out how much alcohol might be in the blood: given 50% of people will die when their blood reaches 0.4% alcohol, that's a good benchmark figure to use. So, if you drink 5 liters of blood, which is 0.4% alcohol, what will happen? Aside from throwing up, not a lot:
[Y]ou would have ingested a total of 2 grams of ethanol, which is the amount you'd get from one shot glass full of beer. Drinking that much blood would, in turn, raise your own blood alcohol level to 0.005. In other words, only 1/60th of the alcohol in their blood would make it into yours.
If, after you drank all this blood, someone killed you and drank your blood, they would then have a BAC of 0.00008. If this process were repeated about 11 times, there would be fewer than 60 molecules of ethanol left in the last person's blood. After one more cycle, there would very possibly be none. They'd just be drinking regular blood.
The short answer then is: no. Also, just so we're clear, it's worth pointing out that drinking blood is a really terrible idea. Don't do it. I won't even get you drunk. [What If?]