Upgrade log could give you some hints. You can find the upgrade log file at :
C:\Program Files (x86)\EPiServer\Shared\Install\Logs

When trying to upgrade, two logs are generated in C:\Program Files (x86)\EPiServer\Shared\Install\Logs:

2012-07-25T14 15 15 EPiServerInstall.exe.log:

EPiServerInstall Information: 0 : Starting in C:\Windows\system32 with "C:\Program Files (x86)\EPiServer\Shared\Install\EPiServerInstall.exe" ,
EPiServerInstall Information: 0 : EPiServerInstall.exe started with arguments "",
EPiServerInstall Information: 0 : Displaying Deployment Center,
EPiServerInstall Information: 0 : Initializing... ,
EPiServerInstall Information: 0 : Popuplating sites... ,
EPiServerInstall Information: 0 : Populating products... ,
EPiServerInstall Information: 0 : Populating actions... ,
EPiServerInstall Information: 0 : Updating compatibility information... ,
EPiServerInstall Information: 0 : Downloading compatibility information...,
EPiServerInstall Information: 0 : Compatibilities are unchanged,
EPiServerInstall Information: 0 : Getting scripts for Search 1.0.517.275,
EPiServerInstall Information: 0 : Getting scripts for RelatePlus 2.0.0.0,
EPiServerInstall Information: 0 : Getting scripts for Mail 5.1.343.270,
EPiServerInstall Information: 0 : Getting scripts for Community 4.1.517.358,
EPiServerInstall Information: 0 : Getting scripts for CMS 6.1.379.0,

2012-07-25T14 15 38 EPiServerInstall.exe.Upgrade Site (SqlServer).log

EPiServerInstall.exe Information: 0 : Database Settings
Database Name : EPiServer_CMS
Database Server Name : .
Site Settings
Site Path : C:\Projects\EPiServer\Web
Site Name : EPiServer CMS
Installer Database Server Access
Password to use when connecting to the database server : 
Login Name to use when connecting to the database server : 
Use Windows Authentication when connecting to the database server : True

Hi, 

I did a quick upgrade test with vanila site to 6R2 and it ran without any error. Could you give it another try ? 

I've tried 4-5 times with the same result. Did you try to upgrade a vanilla CMS6 R1/Relate 2/.NET4 to R2 with success? 

I forgot to mention, before being able to select what site to upgrade in Deployment Center, I had to click "Show all sites", as my R1 site wasn't listed. 

This being a Relate-site, I figured I had to install both Relate and CMS install packages separatly. I got a bit confused reading all the previous forum posts, blog posts and articles referring to installation orders, copy-pasting of assemblies and what not. So I tried installing Relate R2 before CMS R2, but CMS6 R2 was somehow was installed anyway. So I guess I only need the Relate R2 setup? 

Anyways, I'm still stuck regardless of this.

Edit:
When hovering over the CMS6 R1 site in Deployment Center it says "CMS version 6.1.379.0". I'm guessing it could have something to do with the R1 site having hotfixes with newer assembly versions, thus making Deployment Center detect the site as an R2 site.

Edit #2:
The previous discovery lead me to find a solution. For some reason a couple of assemblies with version 6.1.379.0 was laying in my /bin/ folder, Deployment Center did not want to let me upgrade the R1 site. A removal of these assemblies (or in my case, a simple "Rebuild solution" in Visual Studio) fixed the problem. (Later, I found out that replacing the /bin/EPiServer.dll with a 6.0.530 version works equally well).
However, the upgrade process now stops at:

Error - Cannot resolve dependencies for the following module(s)
EPiServer.Web.InitializationModule

... which I'll look more into next.