A skeleton PC is a pretty fun idea. I always like those kooky PC ideas like a ghost haunting a suit of armor or a tiny or diminutive character.
As far as making a skeleton PC character... there are a few rules that you'd have to dismiss or ignore to make it any fun. For example, a skeleton loses skill ranks and can't gain any new ones.
So let's assume for a sec that we can bend the rules a little. How does that look like? The easiest way is probably as follows: Build a level 1 living PC of any race you want but don't apply a class yet. Then apply the Skeleton template (google the d20pfsrd skeleton template)
Alignment: We have to change the character's alignment to neutral evil. You could change this but it also doesn't prevent us from building a PC out of it, so I would leave it.
Type: changes to undead
Armor Class: if you picked a small race you get +1 natural AC. medium race gets +2 natural AC.
At this point we have to make the *very* important decision about if we're going to level up using Skeleton stats or Class stats. I'll do examples for both, but we have to pick one or the other. What this means is that you can build the PC to be like... eventually they would be either a level 5 Skeleton, or a level 5 Fighter (with undead/skeleton type)
Hit Dice: If we use skeleton level progress then 1d8 per level (remember to use cha for bonus hp). If we use class progression then we use the class dice (1d6, 1d8, 1d0, whatever) +charisma for bonus hp.
Saves: if we use the Skeleton as level progression it's 1/3rd HD for fort and ref and 1/2 HD +2 for will. If you use class progression then use whatever the class bonuses are (whatever is on their class table)
defensive abilities: the PC should gain the 5/bludgeoning and cold immunity plus all undead traits.
attacks: gain a 1d4+str claw natural attack and can do full attack action to attack with both claws at full bonus (skeletons are cool).
Abilities: Dex +2 and no con score. (the RAW says they should have no int score either, but I think it would be more fun for your player to actually do stuff, so give him an int score). RAW also says that wis and cha are set to 10. I would consider the only important things here would be the +2 Dex and no con score. Everything else you could disregard.
BAB: Either use skeleton progression of 3/4 HD or use class progression (again, not both)
Skills: RAW says they lose all skill ranks and can't gain any skill ranks. I'd probably ignore this rule and either give 2+int for skeleton progression or use class progression+int.
Feats: RAW says lose all feats and gain improve initiative. I would say to instead use normal feat progression for character levels and if you are using class progression then to give class feat progression. If you want to use skeleton level progression I would just look at Fighter feat class progression and borrow that.
And then there you go, a skeleton PC.
I would personally probably use the class progression method I described above, as there isn't any flavour to leveling up as a skeleton. Moreover if the skeleton is neutral evil (as RAW) then you couldn't be a paladin due to alignment requirements. But again you could disregard the forced alignment change and justify it as 'free will'.
As far as healing and damage. many undead (including skeletons) do not heal naturally from injuries and if brought to 0 hit points are destroyed. So it's fairly risky to play as a skeleton. Plus people will probably hate him. An undead can be healed by negative energy from an evil cleric (or antipaladin lay on hands I guess? I forget how antipaladins work). A potion of inflict light wounds or inflict moderate wounds would work to heal the skeleton if he drank it (while damaging anybody else if they drink it).
An important aside about negative and positive energy. When a good cleric decides to use his channeling ability he has to decide if he is going to use his positive channel to either heal living creatures or damage undead. I think there is a feat that lets him do both at the same time but normally he has to make the choice. So if your skeleton PC became an evil cleric or otherwise was able to channel negative energy, whenever he used the ability he could decide whether to use it to heal undead creatures or damage living creatures (but again, without a feat he couldn't do both at the same time). So he could heal himself during combat OR do AOE damage to any living enemy (and also his nearby friends).
Anyway it's an interesting thought experiment. I hope this rough analysis helps.
