Complex sentence: When her contract with Modern came up in 1960, she decided to sign with Leonard Chess' namesake label, Chess Records, and shortly afterwards got involved in a relationship with singer Harvey Fuqua, founder of the doo-wop group The Moonglows.
Split into simpler sentences: When her contract with Modern came up for renewal in 1960, she decided to sign with Chess Records. Shortly afterwards was involved in a relationship with singer Harvey Fuqua, founder of the doo-wop group the Moonglows.