If you file a tax return for other reasons, you must include a schedule C if you have any amount of self-employment income. If self-employment is your only income, you must file the schedule C if your net profit after expenses is more than $400.
You do not need to form any particular business structure, like a corporation or LLC. There may be advantages to doing that but that is legal advice we are not equipped to give. You may be required to register as a business by your state laws, but that does not change your federal tax situation if you do or don't register.
One other note: you can use your personal SSN for self-employment as long as you don't pay any employees. Once you have employees, you must get a federal EIN (employer ID number). You may want to do this now in any case -- if you have an EIN, you give that number to your clients and customers instead of giving them your personal SSN. You can get an EIN online in a few minutes.
https://www.irs.gov/businesses/small-businesses-self-employed/apply-for-an-employer-identification-n...
