I have a set of self-amplified speakers like these:
... and also a set of poor-condition, original Astone Versa 200 speakers.

After starting to use them recently, I've found that they play music much better if I cut out the lower end with equalisation in my software.

Could I easily build an electronic high-pass filter to incorporate into the speakers (I'm more concerned about the bigger set, not the Astones really), and how would I do it?

Would it be easy to have it adjustable (attenuation, cutoff, filter width/hardness)? Could I add a bass boost?

A high pass filter is typically just a capacitor in series.  For steeper curves, there are simple circuits involving capacitors and inductors.  People building their own speaker or custom car stereo systems do this all the time and there's plenty of info and calculators on the web.

These are usually used with unamplified, low impedance speakers.  If you want to use them at line level, put an approx. 600 Ohm resistor in parallel with your amplified speaker's input and use that as impedance in your calculations (the actual impedance of the amplifier's input is unknow, but likely quite high).  At the line level, you may also have to use shielded inductors (if your circuit requires them) to avoid picking up AC mains frequency buzz from household devices emitting magnetic fields.

Circuits of this type are generally not adjustable.  You can achieve some limited adjustment (for example, by adding a pot of about 10x your speaker impedance in parallel with the capacitor in a first order filter), but it won't substitute for an equalizer.  Bass boost would come from a low pass filter, so it is the opposite of what you are asking for in the rest of your post.  I am not sure how that fits in, unless you have a speaker you want to dedicate as a subwoofer.

Overall, you are not going to get good sound from bad speakers no matter what you do, so other than the educational value, this is going to be a big waste of time.