You can apply for a current account with overdraft as you do for any other current account, by talking to the provider you want to move to. If you're looking to switch your current account over entirely, the Current Account Switch Service should be a great help, as it automatically moves your direct debits and standing orders to your new provider, meaning you don't have to worry about missing payments.
Before you apply for anything, however, remember to check your credit rating. Your credit score at the time of application will greatly affect whether you get the account or not, so if there's anything you can do before applying to improve it, don't hesitate.
Also remember that changing your bank account will likely reduce your credit score slightly, due to the credit check that will be part of the application, so be sure of which account you're after before you apply and don't immediately apply for a different account if you get rejected.
All banks offering overdrafts must now have an online tool on their websites that will allow you to see if you are eligible for their account, before you apply. This allows you to see if it’s worth applying, but without any impact on your credit rating.