Here's how I did it with Ubuntu 14.04LTS. I imagine it might work with Mint, which is based on Ubuntu.
Method 1
I opened a command line terminal (With Ubuntu, it's Ctrl-Alt-t).
I typed in the following commands, hitting <enter> after each line.
sudo add-apt-repository ppa:libreoffice/libreoffice-5-1
sudo apt-get update
After typing in the first line, you'll be prompted for your user password. The first line adds the repository for LO 5.1.x. The second line automatically updates your system to use it.
Method 2
You can do the same thing through Synaptic Package Manager. Open Synaptic and click on Settings and then Repositories. In the Repositories dialog, click on Other Software, then click Add. Then type in ppa:libreoffice/libreoffice-5-1 and click Add Source. You will be prompted to reload your software sources. Once you do that, you can click Mark All Upgrades. Synaptic will mark the LO packages that are to be updated. Click Apply and you'll be upgraded.
Again, I've used both methods with Ubuntu. That said, I am not by any means an experienced Linux user. I tend to use Synaptic for all my software installation needs. I've never downloaded and installed a *.deb file. Instructions for these methods are found at:
https://launchpad.net/~libreoffice/+archive/ubuntu/libreoffice-5-1.
I don't know enough about Mint to know if it will work properly, but since Mint is based on Ubuntu, I tend to believe it will.
Virgil