Firstly...how much storage have they given you for free? Usually it's around 2GB which isn't actually very much. The idea really is to lead you into buying more space which is how they make their money.
What I would recommend instead is Dropbox
They will give you 2GB for free and you basically install an application onto your PC which will create a "Dropbox" folder under your documents. Any files you put into there will automatically get uploaded to the cloud.
So to answer your question of how to put photos on it, you copy/move them into the Dropbox folder and just wait. That's it.
Secondly (and this is a feature I love the best from Dropbox), you can download the Dropbox app for Apple/Android and enable a feature called "Camera Upload". What this will do is automatically upload all the photos
you have taken with your smartphone to the Dropbox folder everytime you connect to wifi.
So for example if you go out for the day, take some family photos. When you return home they will be uploaded automatically without any input from you. If your phone dies/stolen whatever then your photos are still safe on Dropbox!
If 2GB isn't enough you can refer people to Dropbox and get more space for free. Of course you can also pay them!
The biggest downside of something like BT's offering is what happens if you ever decide to get ADSL from someone else? What happens to the stuff you have stored in BT Cloud?