The Blues completed the signing of the Brazil forward from Russian club Anzhi Makhachkala for a fee thought to be £25.5m at the end of last month.
The 25-year-old, who was poised to join Tottenham until Chelsea intervened, penned a five-year contract and will wear the number 22 shirt.
Willian will face stiff competition for a place in the first team, however, with the likes of Eden Hazard, Oscar, Juan Mata, Ramires, Andre Schurrle and Kevin De Bruyne amongst the names vying for a starting role under Jose Mourinho.
"I know that I haven't secured my spot here," he told 'Globo Esporte'. "I'll do my best to achieve that place, while respecting the other players. I'll train, play and do my best to get the trust of the manager and of the fans."
Willian hopes his move to the Premier League will boost his chances of making Brazil's squad for next summer's World Cup.
"There is no doubt English football is very big, with huge visibility," he added. "I hope that helps me to be called up for the national team - but of course that depends on me.
"I'll have to play and prove I deserve that spot. I first have to become a regular starter [with Chelsea], then play well and be called up [by Brazil]. This is my goal."
The 25-year-old spent just eight months in Russia, where he made 15 appearances for Anzhi before jumping at the chance to join Chelsea.
On the move to Stamford Bridge, Willian said: "Things just happened. Chelsea came with an offer and everyone knows that I wanted to play for the club since I was at Shakhtar.
"It shook me but I didn't think twice. As both presidents (Suleyman Kerimov and Roman Abramovich) are friends, that helped too. It was a very easy negotiation.
"The Brazilians called me, especially David Luiz, who told me to come quickly. I was very happy to have them around and they will certainly help me a lot.
"I wanted to play at Chelsea, even more so now with Jose Mourinho, who is the best at what he does.
"When I had a day off, I came to watch the club's games, so I am more than happy for having made this transfer become true."