"I'm excited to bring a championship here to LA in a Clippers uniform," Paul said.
Paul was acquired by the Clippers from the league-owned New Orleans Hornets on Wednesday night.
Los Angeles sent guard Eric Gordon, center Chris Kaman, forward Al-Farouq Aminu and Minnesota's unprotected 2012 first-round pick to the Hornets for Paul. The Clippers also received two future second-round picks to complete the trade.
The 26-year-old Paul requested a trade soon after the five-month lockout came to an end.
But the league nixed deals between Paul and the Lakers, and then the Clippers, before accepting Wednesday's trade.
"It's been an emotional roller coaster. I'm excited to finally be here and get this thing going," Paul said.
Paul and All-Star forward Blake Griffin will potentially form one of the most dynamic tandems in the league.
"In order to be able to wake up from a dream, you have go to sleep. I haven't slept in about a week and a half," Paul said.
A four-time NBA All-Star selection, Paul averaged 15.8 points and 9.8 assists in 80 games for New Orleans last season.
Selected fourth overall by the Hornets in the 2005 NBA Draft, Paul showed early on that he was going to be a star in the league. The Wake Forest product averaged 16.1 points and 7.8 assists to earn NBA Rookie of the Year honors in 2006.
Paul has averaged 18.7 points, 9.9 assists and 2.4 steals over 425 career games with the Hornets. In 23 playoff games, Paul has averaged 21.9 points, 11.1 assists and 5.3 rebounds.
A member of the All-NBA First Team in 2008, Paul was selected to the Second Team in 2009 and Third Team in 2011.
Paul made sure he paid respect to New Orleans.
"There's no way I can talk about the future, and what the future holds, without paying homage to the city of New Orleans, its fans, the employees and my friends and family in the community who have made me what I am," Paul said.