I figured re-porting this might help those in question of how to obtain the achievements for battling the secret characters on the arcade ladder. Best to try these on Beginner difficulty to make things easier.
Reptile - You must fight on the 'Pit 2' stage at night. Wait until you see a shadow of a witch or Santa cross the moon in the background and get a double flawless victory, then perform a stage fatality. It's best to reload the ladder and wait until you get the stage as the first or second battle.
Noob - On the 'Temple' stage, a priest and a corpse are usually seen at the altar in the background. Instead, to face Noob, you must see Noob standing next to the altar and nobody else. He will be surveying the fight. You must win the match without blocking to engage him.
Smoke - Possibly the easiest one. In the 'Living Forest' stage, head to the right to see if Smoke is leaning out from behind a tree. He shouldn't be too hard to miss. If he is, hit D + Back Button/Select (whichever system you're playing on). Your opponent should instantly explode and the fight with Smoke will commence.
Jade - Possibly the hardest one to achieve. You must go through the ladder without losing a single match or round while reaching Shang Tsung. From there, you must double flawless Shang and perform a fatality on him. You cannot lose a single round to him, either. You cannot cheat with having it set to one round wins. Two rounds, at least, have to be set! If done correctly, you will see a new loading screen and not the ladder. Jade will be waiting. A trick to getting this; if you get hit, continue fighting and win against him still. At the "FINISH HIM" screen, instead of delivering the final blow, choose player select and choose who you want to try again. Better than running through the ladder over and over.