The Nosler Ballistic Tip and similar type ammo operates like you're suggesting, sort of. But not for explosion, merely for assured expansion. The tip of the bullet is a plastic wedge that on impact is driven deep into the bullet, causing sure expansion. If this were coupled with a weakened internal structure (as in a scored jacket, for starters) then such a design may fragment well at lower velocities.
Yes, installing a primer in the bullet is illegal. Given the huge acceleration shock of firing, it would be almost certain to go off immediately, but the power level of a primer is low enough that it would not cause significant damage to the bullet, meaning it wouldn't help it to expand or fragment to any usable extent.
A shock-hardened primer over a small explosive charge, though, WOULD work...and be illegal.
Oh, hell, just drill out the bullet, pour in mercury, and tape over the hole with copper tape and solder it in place. (On a bullet that is NOT loaded into a shell!) The victim gets mercury poisoning. Probably also illegal, though.
For the HIGHLY retentive types, collect all the old ball point pens you can. Remove the tiny little metal roller ball from each one and save them up. They're usually made of Osmium, which is MUCH denser than lead or even mercury, and is harder than a tax collector's heart. Save up a whole shotshell full of them...but watch the weight so it doesn't go over the load spec.
They'll carry up awfully well for shot that small!
Fuse any shotshell's shot load into a frangible slug by pouring in candle wax that's just hot enough to flow easily. It makes a wax/shot matrix slug that hangs together pretty well, fragmenting on impact. Pre-warming the shell (but not TOO warm!) will help.
CJ