This took me way too long to get working so here's a guide to what to do.

I tried with apt-get install wine as standard but this version of Wine was old and winetricks kept telling me to remove files because of a sha1 mismatch. So here are the steps to make Path of Exile work on Ubuntu - note: this is pre-Ascendancy though the game is currently working with the new compiler version patch so I can't see why it wouldn't once Ascendancy is released.

First up, use PlayOnLinux because it makes everything easier. This part is fine and will get you most of the way.
sudo apt-get install PlayOnLinux

Once you've installed PlayOnLinux you need to open it up, I had to log out and back in for it to appear. Either use the launcher or PlayOnLinux in a terminal.
Path of Exile in the PlayOnLinux games section.
Once open, hit "Install", select "Games" and find Path of Exile in the list.

Wait. It's not that easy. PlayOnLinux will download and install Wine and run some configurations for you specific to the game, which is neat however there are more tweaks which will need to be added for the game to actually play. At this point you can run the launcher and almost download the entire game.

If you attempt to play now, you'll get errors relating to MSVCP140.dll, which is part of the Visual C++ 2015 Runtime, which isn't an option in winetricks.

Let's get past the Launcher issue. Based on this thread.

Download this file:

Then extract it to the Wine C: drive
unzip -o -d ~/.wine/drive_c/windows/system32/

Now you need to add a registry file to the Wine setup.
regedit ~/.wine/drive_c/windows/system32/vcrun2015.reg

This didn't seem to work for me so I had to copy that file to My Documents, load up PlayOnLinux, click Path of Exile -> Configure -> Wine -> Registry Editor and import the key.

Once you've done that if you click Path of Exile -> Configure -> Wine -> Configure Wine.
Load the Libraries tab, it should look something like this:
Libraires required to override in Wine.

You'll need to add in api-ms-win-crt-time-l1-1-0 to this list too, if you don't the launcher will function but once you begin the game it will crash and not load.

After this, you can now start the game and download the content, once you're done the game should load up and be playable.
Note: Don't click the "Stream" tab. It will crash.