Henderson admitted it had been a hectic week after he was allowed to complete his move to Anfield before linking up with the England Under-21 squad for their UEFA European U21 Championship campaign in Denmark.
The midfielder, who turns 21 on Friday, reportedly cost in the region of £16million and some have questioned whether Liverpool paid over the odds.
But Henderson has vowed to prove he is worth the money and revealed he plans to begin repaying his fee by helping the Reds qualify for the Champions League.
"I don't really make the figures up but it just shows that Liverpool have faith in me," he said.
"So I'll work hard and and improve as a player and learn a lot and hopefully I can repay them."
Henderson has a chance to win over the doubters while on international duty this summer, but ultimately he will be judged on whether he manages to help Liverpool get back into the Champions League, which they have missed out on in the past two seasons.
He added: "Every player wants to play in the Champions League, but I think we've got to focus on the Premier League first.
"Hopefully, this season can be a good season for Liverpool."
Henderson insisted it was not an easy decision to quit Sunderland, the team he grew up supporting.
"You've got to think about these things," he said.
"It's a massive, massive thing in my life and my career but obviously it's a great opportunity for me to go there and improve as a player.
"So it was an opportunity I couldn't really turn down to be honest."
Henderson added of the past five days: "It's been a little bit hectic but it's been brilliant as well.
"So a really good week for me and hopefully it'll get better."
Henderson was unable to show off his skills in England Under-21s' 1-1 Group B draw against Spain on Sunday night, which saw Stuart Pearce's side get out of jail in a game their opponents completely dominated.
"Yesterday, Spain made it really difficult for us," said Henderson.
"They're obviously a gifted team on the ball but I think that we showed great character and a good attitude to stay in the game and get the late goal.
"I think that's what the positives were that we've got to take out of the game.
"We didn't have much of the game but we managed to get something out of it."
There is always the danger England's players could feel they have got the hardest group game out of the way ahead of Wednesday's clash with Ukraine and Sunday's match with the Czech Republic.
Henderson urged caution, though, and said: "Ukraine and Czech Republic are two massive teams still.
"We've got two big games coming up but we'll prepare for them and we'll be ready for them."