Providing the players knew their look, spells, items, and max hp, and probably stats as well, it shouldn't be extremely difficult.
Class
The highest stat a character has should tell you enough about your class to narrow it down to 3 or 4 classes. The secondary stat should then narrow it down to which class you are. (Strength and Charisma for a Paladin or Charisma and Constitution for a sorcerer etc)
If that doesn't do it, the spells and weapons one carries will tell the players their class, providing the characters that they play have logical weapons and items.
Race
Its possible that one can judge their race from what they look like. A few races have telltale signs of that race, so a very small player would be a gnome, or a dwarf or a halfling. It is possible to distinguish between them by knowing how fat they are, and how fast they can move. Distinguishing between elves and humans is pretty easy as all they have to do is feel the shape of their ears, etc etc.
Skills
This is probably the hardest one, and unless they can see their rolls and know the DC, it will be almost impossible to guess at.
Other things
Alignment will be completely impossible unless you tell them every so often that they do not enjoy what they are doing. I hope you will give them their level, but if not they could guess it roughly based on their max hp and the class and the con modifier, and race too if they're a dwarf. Saving throws are easy since its in the very start of the class description.
Basically, all you need to do is answer their questions on what they look like, and give them all standard characters with enough information. The rest can be worked out by themselves. Every so often, the characters might suddenly have a remembering of what they are and decide that they can use a special ability (as you tell the players that they can do that). If you want to be able to tell them their ability without giving too much away, give them some sort of character panic button, and when they press it they instinctively create a random effect (and you only tell the players what visible effects or psychological effects are). Guessing spells is going to be extremely difficult, so at least give them their spell list.
I know the first part didn't directly answer the question, but it should help you to see what can be done with the information you give them, and help you decide what information you can release and when.
Note, darkvision is also a nice way to determine one's race, and if one can't see in the dark and they try to, one might instinctively use the light cantrip (because you decided that they did).