Can I unlock my iPhone on my Macbook?

The end goal is to get my notes backed up onto my computer and then restore this back up onto another iPhone.

You aren’t syncing notes using iCloud?

I am trying to back up my iPhone onto my computer but it won't let me because the phone has a passcode on it

See this -> iOS: Forgot passcode or device disabled

BTW, Recovery Mode restore by-passes and wipes your iPhone password.

You can reset if you want after your screen is fixed, or using a P/W which avoids the part of the keyboard screen that is razzed!

Just get your screen fixed.

IPhone 5 screens are replaced from the top, a fairly simple process, so the chances of losing data on the phone by having screen replaced is minimal.

The only way to by-pass the P/W using iTunes is to do a restore via Recovery mode, but this will erase the phone's contents, and as you say, this is the last thing you want.

On further reflection it appears that your iPhone is currently charged and turned on i.e. the inability to type in the iPhone passcode is not relevant.

If this is so, go to Settings /iCloud/ Backup and Storage now, and back all up to iCloud.

Then plug into iTunes and do a Recovery Mode Restore and select the iCloud option for your contents reload. See the link provided by Chris CA above about how to enter Recovery mode .... just follow it step by step .... easy ..... but be patient and hold the Home button down (sometimes seems a real long time) till the iTunes message "detected an iPhone in Recovery mode" pops up.

You can't get into the Settings of an iPhone that has a Passcode lock (You can try Siri, but that might not work, for security reasons).

And if you get the screen replaced, only do it via Apple or an authorized Apple service provider. If anyone else works on it, your warranty and support are voided.

Unless you set up syncing to iCloud or a mail account, or made a backup of the device to iCloud or iTunes previously then no. If you can't disable the passcode, you can't make a backup to iTunes. I think, however, if you had enabled iCloud backup, then the device would still back up (assuming available space in iCloud) if it is connected to power, connected to wi-fi, and plugged in and charging.

That's one of the security features of the iPhone passcode; to prevent unauthorized access to the contents of the device. If you could make a backup without disabling the passcode (or without the Apple ID/Password for iCloud), then anyone that got your phone could get all of your data without your passcode. Makes for a very poor security system.