There are six hidden bosses in the game. To open them for play, win the championship of their league. A single loss on your way to the title will put you back at the beginning. Normally this is not a problem; however, the secret W3 Super Heavy League will require you to beat every other heavyweight in the game, so a loss along the way can undo a lot of work.
None of the special characters have any inherent advantage over the normal wrestlers. Depending on the kind of match you prefer to play, some of them are actually worse. Joe Bruiser has no grappling or aerial moves at all - punches and kicks only - but he is extremely effective in a TKO match.
DDP - Must win WCW league
Wrath - Must win DOA league
Black Widow - Must win W3 cruiser
Randy Savage - Must win NWO league
Glacier - Must win IUGP league
Joe Bruiser - Must win W3 heavy