Cheika has succeeded fellow Aussie Dan McKellar, who parted company with the Tigers on Saturday after only one season in charge.
The former Australia and Argentina head coach starts work on Monday when pre-season begins, and Leicester will be hoping his arrival will end a lengthy period of upheaval.
Apart from winning the Premiership title in 2022, when current England boss Steve Borthwick was at the helm, success at Welford Road has been elusive: Leicester finished eighth in the league last season.
"I want Leicester competing on as many fronts as we can," Cheika said.
"It's just one step at a time, together, to build the foundations.
"Every player and every team want to win trophies - you aren't playing the game if you don't want to win.
"But there is a difference between wanting to win and doing what's necessary to win. That is what I'll be doing.
"English teams have been well represented in the top tier of Europe for a while now and we want to get ourselves up to that place as well, making sure that Leicester are back to competing for the biggest trophies."
Cheika held talks with Leicester a year ago as part of the recruitment process that led to McKellar's appointment.
The 57-year-old coached Lebanon at the 2021 Rugby League World Cup and also spent time at the Sydney Roosters as an assistant.
His most recent rugby post was guiding Argentina to a fourth-place finish at the 2023 World Cup, while he took Australia to the final in 2015 and the last eight four years later.
At club level he has won the European Cup with Leinster and the Super Rugby title with the NSW Waratahs.
"I wasn't looking at the Premiership and didn't have the desire to coach in it until Leicester came to me," he said.
"But the opportunity to coach at Tigers and lead this group of players turned my head.
"I want this to be my best coaching yet. I want the preparation and the way we lead the team to be at my best level.
"If I can bring my best level, other people will bring their best level and good things will start to happen."
Leicester chief executive Andrea Pinchen admitted the off-season disruption to the coaching team - assistant Dan Palmer also left this week - had been unsatisfactory.
"I accept that the past week is not what fans expected - and nor did we - but the decisions we have made are for the long-term benefit of Leicester," Pinchen said.
"After lengthy, detailed and very honest conversations with Michael in recent days, we are on the same page about what is now necessary to see this club back on top."
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