?I don't see it necessary to be a Khajiit and a vampire. The only reason my character is a vampire is that I wanted her to be a Nord, and have the Night Eye power.
If you are determined to become one, there are some ruins north of the Falkreath sanctuary, called Shriekwind Bastion. It has 2 entrances that both show up on the map, go to the North one. Make your way towards the room just to the right of the big door, and stop sneaking. A vampire in the next room should hear you and attack. You need them to hit you with the Drain Life spell, possibly multiple times, so get out of range or bash them if you don't get infected. Careful you don't die. After you contract Sanguinare Vampiris, kill the vampire, leave and wait 3 days.
Vampirism doesn't help stealth that much. Magic is another matter. Sneak can rise pretty fast and you can use illusion and potions to supplement it if needs be. If that's your only reason I'd recommend leaving until it comes in a quest or possibly if sneaking without it fails you at higher levels
?It's more like vampires get weaker in sunlight, not stronger at night. A vampire in the sunlight has -100% health, stamina, and magicka regeneration, and -15 x (vampire stage) to health, stamina, and magicka. As for the extra sneak, it's kind of a waste with the Sneak perks Muffled Movement and Silence, the fact that Sneak levels very quickly with sneak attacks, and Muffle enchanted boots. I've had no problem playing thief/assassin Khajiit without vampirism.