Lampard won 13 trophies in 13 years and became Chelsea's all-time leading marksmen with 211 goals from midfield during a decorated playing career at Stamford Bridge.
However, the former England midfielder has just one season of managerial experience having led Derby County to the Championship play-off final in May, when they were beaten by Aston Villa.
Lampard faces what he described as the biggest challenge of his career with Chelsea having lost their best player of recent times in Eden Hazard to Real Madrid last month and serving a two-window transfer ban that prevents him from making new signings.
However, he believes Chelsea can remain in the Premier League's top four.
"We want to be playing Champions League football year in, year out for sure. I think that is something that should drive us all on. We want to develop players and be competitive," said Lampard.
"Of course there are variables, we know at the moment there is the transfer ban, we know about the situation. We know Man City and Liverpool pulled away last year and we should be realistic about that, but we should never stop trying as Chelsea to be up there and I think we should be."
Lampard succeeds Maurizio Sarri, who left England to take over at Juventus last month.
Sarri led Chelsea back into the Champions League next season thanks to a third-place finish in the Premier League and also won the Europa League during his sole season in charge.
However, the Italian endured a frosty relationship with the club's fans due to his rigid style of play and reluctance to field some of Chelsea's promising young players.
Lampard's appointment is designed to rectify that with the club keen to stress his "fantastic relationship" with the supporters.
"Frank possesses fantastic knowledge and understanding of the club and last season, he demonstrated he is one of the most talented young coaches in the game," said Chelsea director Marina Granovskaia.
"After 13 years with us as a player, where he became a club legend and our record goalscorer, we believe this is the perfect time for him to return and are delighted he has done so."
Lampard took two Chelsea loanees, Fikayo Tomori and Mason Mount, with him to Derby last season, where they excelled.
His assistant Jody Morris also has extensive knowledge of the Chelsea youth system having been a key figure at the club's academy prior to joining Lampard at Derby last summer.
"What we have is already a very strong squad and we have young players at a really good age that have not just potential, some of them have already showed it in the first team," said Lampard.
However, he faces a baptism of fire with their first two competitive games in charge featuring a trip to Manchester United on the opening weekend of the Premier League season followed by the UEFA Super Cup against European champions Liverpool.