Castlevania 3 is hard as hell (it has been a while, but I can't remember ever beating the Doppelganger without doing the pause/unpause trick), but once you hear that badass jingle from beating Dracula, you're on top of the world.
What I did was get the doppleganger trapped below by jumping on the upper right hand ledge, then spamming Sypha's homing attack until it died. Hence my never beating that stage (and consequently the final level and Dracula) with any other partner lol.
Dracula was actually easier in a lot of ways. Beating Dracula is all about 1. Standing as far back as possible while you whip his first form so you can have room to move out of the way when the flames start to pop up. You really just have to practice dodging the flames for a while to get a feel for it. 2. Getting a feel for how to run under/avoid the dripping slime from his second form while getting in hits with your whip and axe as quickly as you can. If you save up a lot of axes, which you should be able to in the US version (since you re-spawn at the mid-point of the level instead of at the foot of Dracula's tower when you die/start of level after game over) you can basically just stand in the corner and wail on him with axes in his final form until he dies.
At least that's my strategy. Once saw a speedrunner beat Drac in seconds with Sypha's flame (equivalent to a dagger) of all things. But I'm nowhere near that level and don't expect I ever will be.
Also - USE THE AXE TO GRAB THE EXTRA LIFE in the final level. That will save you so much frustration when you have to take off half your lifebar walking through an enemy to get to the door immediately afterward so you don't die on the collapsing floor.