A credit card is not the most important thing in determining your credit history.
You take out different types of credit, they include revolving credit (credit cards), installment loans (car loans, mortgage) and loans (like debt consolidation, etc). All of these accounts are important to your credit rating.
The way that you pay your bills determine your having good or bad credit. Your credit score will increase a small amount with every payment made on time. It takes 24 months of consistent on time payments to establish a good credit rating.
If a bill is paid in cash or not does not matter, rent, utilities, cell phone bills these are all accounts that do not get reported to the credit bureau. Only those accounts that actually report to the credit bureaus will affect your credit rating.