Uber runs a background check before they let people drive for them as an Uber driver and if you have a DUI on your record in the past 7 years you'll be unable to drive for Uber.
If it's been more than 7 years since your last DUI conviction then you may be able to drive for Uber but there's no guarantee.
California has even harsher terms for Uber Drivers so it will be harder to get signed up as an Uber driver in the State of California with a DUI conviction.