When you speak to one of the vampires in Castle Volkihar, they will direct you to Vingalmo for an assignment. Speak with Vingalmo, often on the balcony overlooking the dining hall. Ask how you can help. Vingalmo says he has found a new candidate, ripe for Harkon's Gift. He is sending you to welcome this soul into the vampire family. Valindor of Riften has his ears in places that we'd love to listening in. Joining our family should increase our possibilities in the south. Put his mind to rest, alloy his fear through vampiric seduction and when he is in a pliant condition, embrace him and bite him on the neck.

You can ask Vingalmo what he means by allaying fear through vampiric seduction. Vingalmo explains that our vampire blood grants us power to calm and relax. If you currently lack the gift, forgo feeding for a day and that power will come naturally. Once your thrall is suitably sedated, you will be able to impart the gift without resistance.

You can also ask whether anyone can become a vampire if bitten by one. Vingalmo answers that while the blessed state of undeath can be accidentally conferred (consider the brutish creatures you find wailing in dank hovels in the wild), Only those who can trace their ancestry through Harkon's mortal blood line can faithfully accept the power conferred by his vampiric blood. And among those, it seems, only when the heavens show the correct aspects. Vingalmo has found such a soul and the stars appear cooperative.

Vingalmo also suggest that when you offer the embrace, do so away from prying eyes. Our new sibling will then likely require rest while his blood undergoes the change.

This starts the quest:

New Allegiances
Vingalmo has sent me to Riften where I am to find Valindor and bestow the Gift of Blood upon him. I should first render him pliant through the use of Vampire's Seduction then embrace and bite him on the neck. If I currently lack the power, I should forego feeding for a day as I will gain the ability. I should then return to Vingalmo while Valindor undergoes the change.

Objectives
Turn Valindor into a vampire.

NOTE: The target you get is chosen at random, so you might receive a different victim. For example, you could get Cairine of Markarth, or Haran of Winterhold to be your target. But the rest of the quest is the same.

Fast travel to Riften and set quest as active in your quest log, so that you can track it with the compass. Also check in your powers menu that you have the ability "Vampire's Seduction". If not, then wait a day or two, and it should say "Vampire's Seduction added".  The power description is as follows:

Vampire's Seduction
Creatures and people up to level 10 won't fight for 30 seconds.

Locate Valindor either in his home or wandering around the city. Wait for him to be in a secluded area where no one is watching. Use sneaking if your sneak skill is high enough and he is in the open. After you use Vampire Seduction, approach him and press the pickpocket button, and it will ask you what you want to do. Choose "Feed".

Once you've done that, return to Vingalmo in Castle Volkihar to report your success. Tell him that Valindor has been welcomed to he family. You will get some Potions of Blood as a reward.

Note: If you return to the person whom you transformed, you will find a coffin has appeared in his house, plus some new dialog.

This completes the quest, Seeking Allegiances.