Poyet, who joined the Seville-based club after less than one year in charge of AEK Athens, played for Real Zaragoza from 1990-97.
"I will give everything," he told a news conference. "It's my character."
Betis are 14th in the La Liga standings with 42 points from 37 matches.
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"We will try to have a starring role and win more games than we lose," the 47-year-old said.
"Our aim will be to try to have an identity, improve and not look back," he added.
"I want a team that respects the ball, that controls the game and that has a special character on the pitch, something which has a lot to do with the coach."
Poyet, an aggressive goal-scoring midfielder who played 26 games for Uruguay, won the King's Cup and European Cup Winners' Cup with Zaragoza before spending four years at Chelsea and three at their London rivals Tottenham Hotspur.
His first coaching job was with English club Brighton and Hove Albion between 2009-13 and he moved on to Sunderland before being fired by the Premier League club in 2015.
Poyet announced last month that he would not be staying at AEK.
"There are times in a coach's career that something has to happen so that you feel complete and when (Betis sporting director) Miguel (Torrecilla) called me and offered the opportunity to coach this incredible club, it was a very easy decision," Poyet said.
"When I faced Betis as a player, I always thought it would be fantastic to be a part of this club."
Betis, who are safe from relegation, play their final La Liga game of the season at home to Getafe on Sunday in what will be Juan Merino's last match in charge.
(Editing by Ed Osmond)
