In D&D 5e we have the vampire race.

Now, I've been listening to MRRhexx's D&D lore videos and heard that if a succubus kisses you, you can become a more powerful true vampire. Is this true?

(The video starts from where he talks about succubi turning people into vampires.)

I do have a player character who wants to become a vampire. And I know the Monster Manual has one way. Get bitten, wake up a thrall then drink the blood of the vampire.

But my question then is: If a succubus didn't kill one with her draining kiss, does one turn into a vampire that way?

Vampire is not an official playable PC race*

There is not a playable PC race option for Vampires in official material for 5e.

Becoming a Vampire Spawn NPC is the closest option

The Monster Manual does state the following if you are bitten by a Vampire and die from the bite (my emphasis):

A humanoid slain in this way and then buried in the ground rises the following night as a vampire spawn under the vampire's control.

Becoming a Vampire Spawn changes the character and potentially takes them out of player control, per the "Player Characters as Vampires" sidebar (MM, p. 295):

The game statistics of a player character transformed into a vampire spawn and then a vampire don’t change, except that the character’s Strength, Dexterity, and Constitution scores become 18 if they aren’t higher. In addition, the character gains the vampire’s damage resistances, darkvision, traits, and actions. [...] The character’s alignment becomes lawful evil, and the DM might take control of the character until the vampirism is reversed with a wish spell or the character is killed and brought back to life.

Note: this is not how you list becoming one. I'm not sure your thrall/drink blood comes from, but I haven't been able to find it.

Succubus' kiss does not turn you into a vampire

The stat block of the Succubus in the Monster Manual details the mechanics of the Draining Kiss:

The fiend kisses a creature charmed by it or a willing creature. The target must make a DC 15 Constitution saving throw against this magic, taking 32 (5d10 + 5) psychic damage on a failed save, or half as much damage on a successful one. The target's hit point maximum is reduced by an amount equal to the damage taken. This reduction lasts until the target finishes a long rest. The target dies if this effect reduces its hit point maximum to 0.

This requires the target to have already fallen under the Succubus' Charm and it does not do anything but what it says above and does not and can not turn you into a vampire.

MrRhexx's lore video is not official 5e or WoTC material

While interesting, and possibly accurate, the information in the video is not official. It also doesn't necessarily line up with 5e specific lore. While I'm not going to make a judgement on it's actual veracity, simply not being an official source and that the information you're presenting here doesn't line up with the current documents for 5e, I'd be wary to utilize the information as official and correct for 5e.

*WoTC did release a Plane Shift:Zendikar supplement under their MTG releases that are equivalent to D&D's Unearthed Arcana and it does include the note that the information is usable for 5e. This material has not undergone playtesting, but it does provide an opportunity for a Vampire player race.

The lore that you're describing here is technically D&D lore, but it doesn't cover all of D&D, and it isn't 5th Ed.

The only place it shows up at all is the old Forgotten Realms book Lords of Darkness (AD&D 1st edition, 1988) where a succubus kiss can create True Vampires. It hasn't been mentioned since, and it wouldn't have any influence outside of the Forgotten Realms setting.

Now, it is the case that Forgotten Realms has put in some effort to allow a history that spans all of the editions (with regular cataclysms to explain why the world starts working differently). Technically one could claim that once upon a time that had been the source of Greater Vampires (and, by extension, of the standard vampires we see today). However, that mechanic is no longer in play, suggesting that (barring a house-rule) whatever it was that cause succubi to create vampires stopped working during one cataclysm or another.

Incidentally, there is one other way suggested that one could become a vampire. Strahd von Zarovich, Darklord of Barovia, master of Castle Ravenloft, of the demiplane of Dread is said to have become a vampire as a result of a "pact of blood" with an unknown entity (Strahd states that it was Death itself). This is apocryphal, however, and isn't so much a player option as a suggestion on how a DM might arrange for vampirism, as part of a major plot arc, if they saw fit.

Talk to your GM

There are no official rules for a PC becoming a vampire, but your GM can make it happen if he likes the idea. Make sure that the other players are on board, as this could change the tone of the campaign in a direction they might not want. D&D Beyond offers a homebrew vampire race to save your GM some work.