Seeing so many problems encountered by a few players, I decided today to do that quest.

I spoke with the Innkeeper.
Went to the Jarl.
There was an arrow indicating the ghost of the girl standing just behind the wall (it was day-time).
She asked me to find her after dawn so that she can reveal me a few things.
I went to the graves and saw an open one with a coffin. You can open it but I did NOT as she said at night.
I waited until 10.00 p.m.
Laelette, the Vampire, attacked me and I killed her.
There is an arrow now on the coffin. I open it and Helgi talks to me.
Thonnir arrives and leaves.
I must talk to him.
I must investigate the house of Alva.
I found her journal which must be given to the Jarl. (I was attacked by Hroggar and had to kill him.)
The Jarl asked me to kill the Vampire Movarth in his lair right near.
At the lair, the folks were too frighten and only Thonnir accepted to come with me (he is of no help).
I killed them all and also Alva who attacked me.
I returned to the Jarl to become Thane.
I checked my Journal and there is no trespass, no item stolen, no assault, no murder, in the Crime section.
End..
All that without encountering any problem, bug or glitch. I am on PC. I play a vanilla game which means no mod, no addings. I never use console commands. I only save a lot.

I really hope that this will help!