How could a PA-28 be made to fly higher?
The Turbo Dakota (PA-28-201T) is very close to what you seem to be describing and is likely much less expensive to buy outright vs. the time and materials required to modify a lowly PA-28-1xx.  The published ceiling is FL200 with its TSIO-360 but at light weights* I'm sure a determined pilot could coax it to FL230 or FL250.

So, the answer is: a big TSIO engine.  The 260hp TSIO-470-B (only TSIO-470 made to my knowledge) would probably do the trick.  A TSIO-520 has taken me to FL250 in a T210 so I know it would probably take a Cherokee at least that high assuming that you successfully attach the engine to the plane (approvals, engineering, mechanical, W&amp;B, etc).

*The Dakota is one of the few small single engine planes with a gross weight of more than twice its empty weight leaving plenty of wiggle room for a pilot seeking to extract better than book performance from the plane.

**Answer assumes you have an O2 system that will do the job at FL250.