@theoderic Understand that cheques are simply a way of moving money from one bank account to another. They are functionally identical to handing you cash, sending an eTransfer, or wiring money directly into your account.
Your employer, however, is legally required to withhold certain amounts of your pay and remit it to the government as taxes. And as such they are required to keep records and file forms that tell the government how much they paid you. You get a copy of it as well (this is your T4). This is the official record of your income, and it is up to you to doublecheck that is correct.
If your employer DOESN'T keep track of this and instead just hands you some money with no T4 or pay stubs paperwork... That's called being paid "under the table" and it is illegal. Doesn't matter if it is cheque, cash, or and electronic funds transfer.
"Doing your taxes" is when you submit your T4s and other forms at the end of the year to the government, calculating your taxes, and seeing if your employer sent too much money or not enough. This is your opportunity to correct anything wrong and claim any additional benefits you are entitled to which might make you have to pay less taxes. You settle up any difference when you file (either you get a refund, or you have taxes owing you must pay).
You don't do these math calculations yourself... You either use a computer software like WealthSimple to do it for you, or you hire an accountant to do it for you.