The Portuguese has signed a four-year contract and returns to the club he led to two Premier League titles, and FA Cup and two League Cups between 2004 and 2007.
"I never hide that in football I have two great passions - Inter Milan and Chelsea - and Chelsea is more than important for me," he told Chelsea TV, before adding a message to the club's fans.
"Now I promise the same things I promised in 2004 with this difference to add - that I am one of you.
"Now we are back together at a great moment for both. We are to ready to marry again and to be happy and successful again.
"I feel myself not just a Chelsea manager, it's the first time in my career that I arrive at a club that I already love.
"I am happy and proud because when you return to a place it's for some reason, it's not because you had good results in the past, not just because the fans like you, it's because you left something here. I left something here."
After falling out with Chelsea's billionaire owner Roman Abramovich, Mourinho moved on to Inter Milan, where he won a second Champions League after a first with Porto, and then to Real.
"Knowing these different football cultures, being in different countries, was important for my evolution as a manager and as a coach," he said.
"I think I arrive in my club, my Chelsea, in the best moment of my career, my stability as a manager, my stability as a person.
"I arrive in the best moment ready to give everything I have to try and make the club happy.
"I'm the same nature but much more mature with a different approach to things. Much more ready to establish myself in the club and stay for a long time."
Mourinho, 50, said he would not live off past successes at the club.
"I'm very demanding with myself. I want to start from ground zero. I want to work hard again, to build a different team from the one I did in the past. I want to have that pressure on myself, like it was the first time I was here... make people feel they have a reason to be happy with me," he added.
Rather than spend millions of pounds on big-money transfers, Mourinho said his priority was to "improve the team through work" before considering any potential signings.
"If I don't do that I'm not happy with myself. If after that we can improve the team by buying a couple of players it would of course be fantastic," he said.
The Portuguese remains a hugely popular figure at Stamford Bridge.
His return will be greeted with delight by Chelsea fans, who protested bitterly against his predecessor Rafa Benitez.
Chelsea's chief executive Ron Gourlay said: "I am delighted to welcome Jose back to Chelsea. His continued success, drive and ambition made him the outstanding candidate.
"It is our aim to keep the club moving forward to achieve greater success in the future and Jose is our number one choice as we believe he is the right manager to do just that.
"He was and remains a hugely popular figure at the club and everyone here looks forward to working with him again."
Mourinho ended his three-year tenure at Real Madrid on Saturday with a 4-2 win over Osasuna.
Although he won a Liga title, he had a turbulent relationship with senior players such as Iker Casillas and lost the support of many fans.
Mourinho will be accompanied by three staff - Rui Faria, Silvino Louro and Jose Morais - as assistant first team coaches working alongside current first team staff Steve Holland, Christophe Lollichon and Chris Jones.
Chelsea said he will be officially presented in a news conference at Stamford Bridge on Monday, June 10.