Stellini, who stood in for several games earlier this season while Conte was recovering from gallbladder surgery, took over from the Italian until the end of the campaign - with Tottenham in fourth place in the league.
"I want to clarify that the decision was taken for the best of the club and for everyone," Stellini told reporters ahead of Monday's trip to Everton.
"It's an important job, I feel responsible. But I have to say I'm ready to do it and I feel I can do it.
"My relationship with Antonio is still the same, very close and nothing changed between us. I have to say thank you because I have improved a lot in my career under him and it's because of him."
Conte's departure came in the wake of his furious reaction to a 3-3 draw with Southampton in the Premier League -- after which he lambasted his players.
"Antonio said what he felt in that moment and of course we were not happy in the end of the game in Southampton," Stellini said.
"Everyone was unhappy in that moment. All the team felt disappointed in that moment."
When asked if his stint at Spurs could be a step towards becoming a first-team manager, Stellini said: "it's the next step to work hard for the next 10 games. Nothing has to change in my mind. I'm not here for vanity. I'm here to help this club.
"I'm very, very focused on the next 10 matches. It's not important what I have in my mind (about getting the job permanently).
"I have to be myself, like I said when I was in charge when Antonio had the surgery, I use my character to manage the team. We'll see what happens during the process. We've not had many days with all the team."
Stellini added that Spurs could have captain and goalkeeper Hugo Lloris back in goal against Everton after his lengthy lay-off due to a knee ligament injury.
"Hugo has been training since last week. Training with the team, we didn't have more of the team until Wednesday," Stellini said.
"Hugo is fitting in well. We have to take a decision with the keeper."
