Yes, replace the speakers. You can drop money in for a new HU, but it will sound almost identical unless you put some better speakers in. Even if you don't replace the HU, you can still hear a difference with better speakers.
As for wiring it, yes you can. To run it off the stock HU you need a line level adapter. Splice the leads into two sets of speaker wires and plug your RCAs into the box. Done.
You can also put in a new HU and it might work. If it doesn't you would need to bypass the amp.
<span style='color:gray'><span style='font-family:Courier'>1995 e36 3 Series RIP
1999 e46 3 Series BURN IN HELL
2000 MR-S Roadster= Current</span></span>