Why did he kill this Felassan?
Felassan was a "Dalish" elf that befriended and mentored Briala during the Orlesian Civil War. I say "Dalish" because it's all but confirmed that he was an ancient elf working for Solas (He had vague origins, knowledge about elfy stuff, made a lot of comments about Elvhenan, had a lot of stories about Fen'Harel, worked for a mysterious entity that he met with while dreaming in the Fade.) His leader had sent him to get the eluvian password from Briala so that his agents could use the Crossroads, but after getting close to Briala, Felassan changed his mind about getting her password and knowingly turned it down before she could give it to him. So he meets with his unnamed leader in a dream, confesses to his failure, and is promptly killed by the leader.
Speculation varies on Solas' exact motivations. You'll hear everything from "He didn't want to kill Felassan but he had to because Felassan had willing betrayed Solas and could compromise the entire plan" to "Solas is a dictatorial maniac who'll straight up murder you for simply disagreeing with him."
Opinions really run the whole gamut on this guy.