When I changed mine (last name, but the process was the same for first), I went to the provincial website, found the number for the provincial statistics office, called, they mailed me a form, I filled it out and had it notarized, provided them my passport and drivers license and then got a certificate in the mail a few weeks later. I then took that certificate to get my DL, passport, SIN (your SSN), etc changed. I also scanned it so I could email it to places like paypal, airmiles, etc. to change my name on those.
Name changes are a matter of public record, so if you're planning on using it to hide someone, it may not work if the person looking for them is smart enough to notice that they've literally disappeared.
The name changee may also run into creditors and have to convince the creditor that they are not the person who owes money, merely someone who changed their name. That was a hilarious phone call.