Villa are sitting in 13th position in the Premier League standings, seven points clear of the relegation zone with six games of the campaign remaining.
Lambert's side have lost six of their last nine league matches, with two wins and a draw since the end of January, and head to Crystal Palace on Saturday in desperate need of points.
Since the Scot joined the club in the summer of 2012, he has had to work with a shoestring budget, with chairman Randy Lerner tightening the purse strings at the club.
He said in The Sun: "Am I happy where we are? No!
"Did I come to Aston Villa to do this? No! This club should be higher. It's got a great tradition and a great pedigree.
"We've got to improve. I knew the first few years would be really hard because of what we had to do. But we have to get this club up there.
"I'm certainly not sitting here thinking this is good. We have got to do everything we can to get this club up the ladder.
"There is a good nucleus of younger players. That is an area of major progress.
"But it's like turning an oil tanker around. It's a long-term project."