The quest begins when Geralt finds an arrest warrant for the Professor on the Notice board outside the Outskirts Inn and ends with the Professor's death in the bowels of the Salamandra base. There is no guesswork in this quest; all advancements are a result of other (often previously-unrelated) quests.
In the phase about the reward offered for Salamander brooches, despite having already met Declan Leuvaarden in Chapter I, you will not be able to hand in the badges until you get the information from the 'Messenger' at the Hairy Bear Inn, which confirms that Leuvaarden is indeed the person paying for the badges. To get this information from the messenger, drink with him (he is a lightweight; you will only need five beers). Don't threaten him or try to bribe him, neither will get you anywhere.
The Professor Edit
| If you find the arrest warrant:
I found an arrest warrant for the Professor — the man responsible for the assault on Kaer Morhen and for Leo's death. If the Professor is a wanted man, Salamandra must have enemies. Any enemy of my enemy is my friend... I should talk to the city guards in Vizima, see if they can't help me contact the Professor's foes. Maybe I can find the Professor's enemies by speaking to some of the city guards in Vizima.
| If you don't find the arrest warrant:
I saw them release a criminal, the Professor, from the city jail... That bastard is responsible for the assault on Kaer Morhen and for Leo's death. If the Professor was arrested, Salamandra must have enemies. Any enemy of my enemy is my friend... I should talk to the jailer Jethro. Maybe he can help me contact the Professor's foes. Maybe I can find the Professor's enemies by speaking to some of the city guards in Vizima.
A Reward for Brooches Edit
It seems a mysterious, wealthy man is offering a reward for Salamandra brooches. Clearly, the gang got on his bad side. Jethro told me I can find this individual's messenger every evening at the inn. I should talk to him. I should speak to the messenger if the man who rewards people for killing Salamanders. I'll meet him tonight at the Hairy Bear.
The Messenger or Working with Leuvaarden Edit
| Talk to the messenger first:
The Messenger Edit
As it turns out, the messenger is working for Leuvaarden. I should speak to the merchant. They say he can usually be found on the Dike. I may be about to gain an ally in my fight against Salamandra... I should speak with Leuvaarden, who can usually to be found on the Dike. (500 XP)
The Brooches Edit
Leuvaarden is cautious. I should convince him that Salamandra is my enemy, too. Maybe then he'll be frank with me. I must bring him three Salamander brooches. The easiest way to get them is by helping their owners leave this world... I should bring Leuvaarden three Salamander brooches.
The Brooches Edit
I have the brooches I promised Leuvaarden, collected off of Salamander bandits. I should see Leuvaarden and collect my reward.
A Reward for the Brooches Edit
I brought Leuvaarden the brooches. It seems the merchant is a fierce enemy of Salamandra. He promised me a nice sum if I dismantle the organization. I suspect he may not be acting alone, but I don't really care as long as he helps. I'll wait for his next move. Leuvaarden offered to help me deal with Salamandra. I await his next move. (600 + 500 XP) or
| Talk to Leuvaarden first:
Working with Leuvaarden Edit
It seems Leuvaarden issued the warrant for the Professor's arrest. The merchant promised me a nice sum if I deal with Salamandra. I suspect he may not be acting alone, but that doesn't matter as long as he helps. I'll wait for Leuvaarden's next move. Leuvaarden offered to help me deal with Salamandra. I await his next move. (600 + 500 XP)
The Professor's Death Edit
The Professor is dead. Two options: dead or alive. I chose the former. This hardly ends my battle against Salamandra, though... Two options: dead or alive. I chose the former. The Professor is gone. (1500 XP)