The German double winners were unable to progress from the group stage last season, but the 23-year-old is hopeful that they can put in a better account of themselves next year
The German champions failed to negotiate a tricky group, which included Arsenal, Olympique de Marseille and Olympiakos, with a matchday six loss against the French side leaving them bottom of the pool.
However, Dortmund were able to regain their form before the end of the season and retain their Bundesliga crown, and the 23-year-old wants his side to take their chance to improve on their previous continental showing.
"We certainly want to get more points [this year]," he told Bild. "And, of course, it is essential to get to the second round! We want to upset the top clubs. We are ready now."
Jurgen Klopp's side also managed to win the DFB-Pokal with a 5-2 demolition of Bayern Munich in the final, and Grosskreutz went on to admit that last season's domestic double felt surreal.
"When I see the pictures, I still get goosebumps. This is unique - cup winners and champions with 81 points. The season was historic."
Dortmund lost star playmaker Shinji Kagawa to Manchester United this summer, but have added Borussia Monchengladbach ace Marco Reus to their ranks.
Reflecting on his side's moves in the transfer market, Grosskreutz is looking forward to the Germany international's impact, but is not fearing for his place in the team.
"Unfortunately, we lost Kagawa. But I'm also excited about Marco. He fits us perfectly and is a huge gain," he added.
"I have now played for three years, and I want to play again and I'll do everything I can. I want to pay Jurgen Klopp back for everything."
The former Rot Weiss Ahlen man has represented Germany on three occasions, but was unable to win a place in Joachim Low's squad for Euro 2012.
However, he is hoping that he can get his international career back on track in the near future.
"Of course I'm hoping for a call up. I have to show what I can do on the field, and hopefully everything will be rewarded."