The artworks have been loaned by Yoko Ono
The display includes artworks created by the former Beatle between 1968 and his death in 1980, and documents key moments in his life.
Many of the 150 pieces depict Lennon's relationship with Yoko Ono and their son Sean, the couple's "bed in" and include a number of self-portraits.
"The pictures show John's love for his family and attitude towards everyday life," exhibition organiser, Jonathan Poole says.
"They are amusing. They are meant to be funny and they shouldn't be taken too seriously."
The main attraction is a collection of Lennon's nine small line drawings, called Multiple Self Portrait.
A picture about meeting a fan in the street, I'm One Of Your Biggest Fans, was drawn one year before he was killed by a fan in New York.
Also included is the Real Love series, which comprises 16 prints drawn for his and Ono's son, Sean.
"This is a major exhibition of John's work and the biggest show in this country for more than 10 years," Mr Poole said.
The pictures have been loaned to the gallery by Yoko Ono.
Animal pictures Lennon created for Sean, also on display, have provided the inspiration for a range of children's products, which will go on sale next year.
Products ranging from high chairs to stationery will bear the images of the colourful creatures.
Two prints will be auctioned to raise money for a children's charity on Friday.
The proceeds of the sale will go to the Rainbow Trust, which provides care for youngsters with serious illnesses.
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Fans can also buy prints at the west London gallery, with prices ranging from £450 to £4,000.
The pictures are on show at the Harbour Yard, in Chelsea Harbour, until Sunday, 9 September.
An image of a naked Lennon and Ono will feature in an advertisement for vodka company Absolut, it has been announced, while officials in the Serbian city of Novi Sad have named a street after John Lennon.