Do you have to be licensed to be a basic at home PC handyman?

You don't need to be a company to be a PC handyman.

Whether you can call yourself a company depends on the state you are in. e. g. in colorado anybody can call themself a company by saying dba (doing business as), but most of the states are not this liberal. But I guess you can call yourself a PC repair business. But as long as you are honest and run a good business you won't be in any trouble.

Where are you mate, Australia? If so then you don't require a trade licence to work as a IT/C technician. You will probably need to apply for an ABN even if you are going to establish yourself as a sole trader you would classify as a business for taxation purposes, so make sure to check this at the ATO website- http://www.ato.gov.au/businesses/ and if you are going to involved in any basic electrical work (not uncommon) you must be certified appropriately, usually a Restricted Electrical Licence is all you will need and it's a good thing to have anyway, look to your state government's requirements- https://www.google.com.au/search?q=restricted+elec...

Best of luck to you