Jurgen Klopp's men beat Napoli 1-0 on Tuesday, thanks to a superb Mohamed Salah goal and a brilliant injury-time Alisson save, to seal their progression to the last 16.
Liverpool finished second in Group C, behind French giants PSG, meaning they are guaranteed to play a group winner in the next round.
Van Dijk, though, is undeterred and hopes that his side can go one step further than 2017-18, when they were beaten by Real Madrid in the final.
Asked about challenging domestically and in Europe, Van Dijk said: "It’s something that we all want, it’s something that we are going to give everything for," he told reporters after their victory.
"We want to play on the highest level and the Champions League, other than the Premier League for us, is the highest level. We want to reach the maximum and hopefully do better than we did last year in the Champions League.
"You need to have dreams, you need to go for it, otherwise why would you be a footballer?"
Van Dijk says he is even willing to castigate his team-mates in a bid to improve the team as a whole, with the Dutchman seen to be barking at Andy Robertson during Tuesday's victory over Napoli.
"You have to, you need to be hard with each other," he insisted. "When things aren’t going right you need to tell each other, otherwise it’s going to be too easy.
"I think that’s a sign of a good team. When people are shouting at me I take it because they want to make me better and their mistake they take it as well if I shout to them.
"That’s how it is, we’re all grown-ups, we take it and after the game if you’re not happy with it then you discuss it. It is what it is."
And in qualifying for the last 16, Van Dijk feels his side showed their quality in the home matches and feels they could have scored more against the side currently second in Serie A.
"I think it was a bit of a classic performance. In the Champions League, in the home games we played, we showed our class.
"Big chances, a lot of pressure, a lot of great moments, and today as well. Like I said, they needed one goal so it could fall [apart], but in the end we did well and we got the job done.
"I understand [nerves in the crowd] because if they score one goal we need to score another two. We had big chances and you’re going to be frustrated when you miss those chances, including myself.
"We could have made it a lot easier, we could have scored another two/three, we didn’t, and then it was [important] for us to win our battles and keep it tight and we did.
Liverpool's week shows no sign of slowing down, with a clash with Manchester United looming at the weekend.
Van Dijk, though, is confident that they have what it takes to pick up yet another victory when Jose Mourinho's side roll into Anfield.
"It’s a great time to be a Liverpool fan or player," he said.
"It’s tough but enjoy it and embrace it - these are the days you want to experience as a footballer and we are."