How to backup an iPhone when the screen is broken, the home button is broken, and it says "Unlock iPhone to use accessories?"
There is no way to back an iPhone up to a Mac or PC with iTunes, or to initiate an iCloud backup, without a working screen to input the lock code, unless you have used the iPhone with the same computer and iTunes before so it is already trusted.
Your best bet might be to take your phone and computer to a 3rd party Apple device service shop and ask if they can please attach a used screen with working touch for a bit so you can do the backup. They might be willing to do that for a few pieces of cheddar cheese...