Liverpool got their Premier League campaign off to a perfect start with a 4-0 victory over West Ham on Sunday.
It means the Reds are the early table-toppers thanks to a brace from Sadio Mane and goals from Mohamed Salah and Daniel Sturridge.
Many have tipped Liverpool to be the closest rivals to last season’s champions Manchester City throughout this term.
City also started the new campaign brightly as they were 2-0 victors at Arsenal thanks to strikes from Raheem Sterling and Bernardo Silva.
Liverpool took their bright pre-season form into the new Premier League year but Klopp knows there is still room for improvement for the rest of the season.
“We have expectations as well, so that is already enough to feel completely different before a Premier League game compared to before all the pre-season games,” Klopp said.
“Wherever they are, whether it’s Man United in front of 100,000, it feels different to a Premier League game against West Ham, that’s how it is.
“You cannot create that situation, that’s what I mean when I said you cannot test the test before you have the test.
“It was good, we know about our expectations and we obviously realised a very positive atmosphere around us in the whole pre-season, that’s all.
“It’s good that the people are positive, but the people know as well that this season will be unbelievably hard.
“That’s the only thing that I wanted to say: we cannot be anything different from the challenger because we didn’t win anything since I am in anyway, [and] it’s a long time ago.
“So we have to invest more, we have to fight more, we have to create more. That’s what we expect from ourselves and then we will see where it leads to.”
Liverpool brought in four new arrivals during the summer, totalling more than £150m.
Xherdan Shaqiri, Alisson, Naby Keita and Fabinho all joined the Reds as they look to challenge for the Premier League title.
Keita shone against West Ham and Klopp was pleased to see his new recruit find his form immediately.
“A good performance, of course,” Klopp said.
“Naby is in because he is a really good footballer and from the beginning he has settled really quick.
“Similar style of play, similar positions as he had at Leipzig.
“I’m pleased, absolutely, but that’s what we expect.”
Liverpool’s next game sees them face Crystal Palace next Monday (8pm).