If you want a simple way of doing this, there is a way of making custom installs using PlayOnLinux that lets you set up wine drives without doing the dirty work.
This worked first try for me(Ubuntu 15.10)
Also, here is a screenshot of it working as a tile in the i3 window manager. osu! insists on still rendering for the entire screen, but it cuts in a way that doesn't put an offset on the mouse input vs graphics, so it's fine for me. http://puu.sh/m5rff/a075dcbb55.png
This worked first try for me(Ubuntu 15.10)
1. Open Configure from the main PlayOnLinux window
2. Click the New button at the bottom left of the window
3. Click next on the wizard that pops up, and select a 32 bits installation. I have not tried 64 bits, but it might as well work.
4. Select wine version. I used 1.7.52, because Leauge of Legends(also wine'd) was using it, so it was convenient. You can mix and match here, but for this small guides sake, let's pretend you need to use 1.7.52. If you do not have this version installed, see below.
5. Name the drive whatever you want. Wait for the new drive to be created
6. Select the newly created drive in the left side of the configuration window, and navigate to the Install components tab. Find dotnet45 on the list, and install it. In some distributions, you need to apply this fix before doing so: https://www.playonlinux.com/en/topic-10534-Regarding_ptrace_scope_fatal_error.html
7. Download the osu installer as you would on windows
8. Back in the configuration window, navigate to the Miscellaneous tab, and click Run a .exe file in this virtual drive. Select the osu installer, and run it. There might be some errors, but for me it turned out to work anyways.
9. (Optionally) Navigate back to the General tab, and click Make a new shortcut from this virtual drive. Select osu!.exe from the list, or browse to the binary yourself.
1. In the main PlayOnLinux window, select Manage wine versions from the Tools menu item at the top of the window
2. Make sure the x86 tab is selected
3. Select the required wine version from the list on the left, and use the > arrow to add it to the list of installed versions.
Also, here is a screenshot of it working as a tile in the i3 window manager. osu! insists on still rendering for the entire screen, but it cuts in a way that doesn't put an offset on the mouse input vs graphics, so it's fine for me. http://puu.sh/m5rff/a075dcbb55.png