Kenny Dalglish's side ended their six-year trophy drought with a victory over Cardiff City at Wembley on Sunday, but penalties were required to separate the sides when the match ended 2-2 after 30 minutes of extra time.
"It's not the end of the season for us, this club is pushing the players day in day out to perform at the highest level." Bellamy told the Liverpool Daily Post.
"That's what we have to continue to try and do.
"The Champions League is the place where this club deserves to be, but we've got to go and prove that on a weekly basis and hopefully we're able to do it."
"It was just a difficult game full stop. I thought first half we started well. We hit the bar and I thought it was a matter of time.
"I don't know how it happened but Cardiff went and scored.
"I couldn't really see it happening at the time but credit to them."
Bellamy enjoyed a loan spell with Cardiff City last season and insists the Championship side were worthy opponents in the final.
"For me personally it's great to win and great to win a trophy but Cardiff were exceptional and credit to them," Bellamy admitted.
"I can only imagine the disappointment they had but they have to hold their heads up high.
"They've done a lot for the whole of Wales."
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