You could run the 3 ways off the HU, but to drive them to their best, you would ideally have to amp them. The internal amplifiers of factory HU's are not the best, especially when it comes to producing a clean quality sound. They have a lot of post processing present in the signal.
At the price point you are talking about, you will struggle to get a good 3 way system with a decent amplifier. You could do it in steps, the speakers first and then the amplifier afterwards, I would however recommend doing it all in one go.
For a good set of 3 ways, like the Hertz ESK 163 L's, you would be looking at around £220 minimum. This is the mid range Hertz speaker. You would them also need the wiring done correctly if you didn't want to cut the factory wiring.
Hope this helps?