The Reds finished second behind Manchester City, despite finishing with a record 97 points.
Henderson said: "A club of this history and magnitude should be challenging for major honours year in year out, and we hope and believe that this is just the start.
"We are all hungry for more. We should look back at our Premier League campaign with great pride too.
"To have achieved 97 points and to only lose one league game in the entire season was a phenomenal achievement by the team.
"Of course, we were bitterly disappointed to narrowly miss out on the title by just a single point, but we will use the huge number of positives from the campaign to fuel us for the new season ahead.
"We will be doing all we can to reward our fans by going one better next season.
"There is something uniquely special about our club and the support that you give us at Anfield that is not replicated anywhere else.
"On behalf of all of the playing squad, I'd like to take this opportunity to thank you, our fans, for always believing in us and supporting us.
"We are very lucky to have the best fans in the world here at Liverpool, and your support really does make the difference."