Yes that should be possible, you need a decent camera, if you don't want it permanently wired to the computer you need an IP camera. You could live streem to youtube, this is also possible.. Or life stream to your personal IP via https.
But than you need to figure a few things out, nowadays with all the wifi stuff we basicaly all have a router/modem. A modem can be approached directly from the www. with your ISP adress. In case of a router you need to set this up with the correct router setting e.g. port forwarding to the correct internal IP adress. Because a router always creates an internal network behind your public address.
Example: Your ISP gives you an IP address something like 82.13.344.155.. So if you are on the www somewhere else you need to fill in that number in the browser to get to your modem. Behind your modem you have a computer or 2 computers that each get an internal address from the router behind the modem. 198.167.0.2 and 198.167.0.3 for the other. Than you have to set to the router, if an external request for entry comes at port 8080 you need to route this reqeust to 198.167.0.2, if you set nothing the router doesn't know what to do and it is dropped. No entrance. But set correctly you reach the port 8080 web interface of the camera set on the specified computer. Or depending maybe directly to the camera if it has its own IP in the network.
Easier for you would be to stream it externaly, youtube for instance or other video streaming hosts, payd or free that's your choice.
But that's likely les private especialy if free.
