If you have a DUI on your record and it's been over 7 years since the DUI conviction and you have no other DUI or moving violations on your record then you could possibly get signed on as a Lyft driver.
Lyft will perform a background check and if it hasn't been over 7 years since the DUI conviction then you won't be able to be a Lyft driver.
California has much stricter laws and it will be even harder to get signed on as a Lyft or even Uber driver with any DUI convictions on your driving record.