Hi guys,
 
I've been using a laptop for years, but my current one is about to die. And I decided not to go with another laptop, but rather a desktop. Mostly because I want to be able to upgrade as the years pass, and because I'm on a tight budget. I would have about 750 Euro to spend.
 
My typical project would look like:
4x Vocals
6x Guitar with Amplitube
1x Bass with Amplitube
1x BFD/Addictive Drums (multitrack)
2-6 VST Sampler + Synth
 
What kind of setup should I choose with this budget. And where to focus? CPU / memory / harddisk speed?
 
Thanks!

16GB of memory, CPU speed as fast as you can afford and an audio drive of at least 7200 RPM (most internal desktop HDs already deliver this).  If you have some extra cash get an SSD 256G or 500G for a system drive.
 
Your system will be future proof for many years.

You will probably do fine with any current generation desktop, there are less performance and latency gotchas than when picking out a laptop.   Your budget should get you a decent configuration.
 
Dave just gave some great advice.  I cannot believe how much performance boost I got just upgrading to SSD for my system drive!  Future proof for years!  So yes, a big SSD for system, and a big 7200rpm secondary HDD for audio.
 
Also, ideally a system should have both USB2 and USB3 ports. You might have some older gear that just doesn't like USB3 ports, so it's good to have options.
 
And I feel that, at least with Intel, that the integrated HD graphics is good enough for everything except serious gaming.  I'm running dual monitors off my integrated video adapter, no problems.  So save some money on the graphics card and spend it on more CPU power, and the other good stuff. :-)

Do not forget about the noise... Easy to overseen when switching from a notebook. Typical desktop has at least 2 coolers (more common 4...), PSU, CPU, Graphics, Case. Depending on you noise tolerance, keeping it "silent" can take significant part of your budget ( real silent/close to silent PSUs are over 100 euro, close to silent CPU cooling is again around 100 euro). Some people have no problem with 30 euro PSU and stock coolers, so up to you to decide.
 
The graphic card is the second to thing about. Build-in is fine as long as you do not need any 3D. The next option is something without fan for under 50 euro. You can buy a card which will be absolute silent in the DAW but provide power when required, unfortunately such cards are hard to find under 150 euro.
 
SSD makes huge difference in many situations. Assuming you make backups, the difference in performance between most expensive and cheapest SSDs is not big (far less then for HDDs or USB sticks). But if you need big disk (>512GB), first select other components. A disk is easy to add later then changing motherboard...
 
The motherboard should not be the cheapest "no name", that can influence the system latency making the whole computer unusable for music.  Choose based on socket/connectors you need (and then check the Internet reviews your choice is not problematic in general).
 
I agree that 16GB of RAM is a "sweet spot" today.
 
So at that point you will just have to decide about concrete CPU model and either you want SSD. And you know how much money still there. So Pentium, i5 or i7, depending from your noise tolerance 

Do you plan to buy an off the shelf computer, or assemble one yourself?  The original post did not mention which path to take, and the advice could differ based on the choice. 
 
A custom built DAW PC with high spec components is probably double the stated budget.  Building your own system will allow you to make the choices that azslow3 stated above, and probably stay on budget.
 
If you are buying an off the shelf system for DAW use, you should at least post the make and models that you are considering to get some honest opinions and feedback.  Shop carefully to avoid potential DAW issues. 
 
Also, it is harder to customize and upgrade the system later if you do not opt for a standard form factor case, power supply, and motherboard from the beginning.  Most OEM's use customized configurations, rather than standard form factors, so a motherboard or power supply swap could be a major challenge later.
 
As far as GPU's go, using on-board graphics can remove one fan from the noise equation.  A fanless GPU can be found, but most of them are on low end versions of GPU's that often don't benchmark much better than the on-board graphics.  They are targeted at home theater PC users that need silent PC's and the latest HDMI/HDCP versions, etc., rather than 3D gamers.  Most current desktop motherboards should offer the connectors for multiple monitors with on-board graphics.