Chelsea completed the signing of the Spain international from La Liga champions Atletico Madrid on a five-year deal last month.
The 25-year-old was top scorer for Diego Simeone's men last term and his penalty in the Champions League semi-final second-leg tie at Stamford Bridge helped Atletico to a 3-1 win on the night and on aggregate.
Having enjoyed success with Atletico, Costa is aware he has to prove himself in a new environment.
"Now I can say I'm really sorry for the part I played in the game we won at Stamford Bridge," Costa is quoted as saying by the British media.
"I remember the atmosphere and the passion of the supporters. They stayed right until the end, getting behind their team, even though it was unlikely they could win after we scored two quick goals. I liked that.
"I hope my actions on the pitch and my feet will do the talking. All I can say is I love playing hard and fairly. I want to improve, not only as a player but as an individual, and I believe this is the perfect place for me to do that
"Joining was an easy decision for me. We have some of the best players in the world, the best manager in the world and it's a club with great history. That's the reason I chose Chelsea and why I want to defend this shirt."
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