If this is for standard Charizard
Both moves are alike and have their own effectiveness against particular enemies and are good moves, but most of the times you'll face Psychic and Ghost types more than dragon types, so in terms of majority Shadow claw will cover you more times. Especially if the region in particular is Kanto or Johto. Also, Dragon type moves will cover only one type, a Ghost type move will cover two, so, if your intention is to cover more Ground, Shadow Claw would be the better choice.
If this is for Mega Charizard X
Mega Charizard X will receive a bonus from STAB and Tough Claws using Dragon Claw, while Shadow Claw will only receive a bonus from Tough Claws. In the case of Mega Charizard X, Dragon Claw would benefit better from its ability, even if you don't find many Dragon type Pokemon to use it against, it will still be an good asset to have. (Also in later generations, five, six and seven, there are more dragon types than the other regions)
So in the end, both moves are almost in the same boat. You should try to compare them basing on what you want to do, if this Charizard is to do something in particular, or if is to be used by standard and also, with region and generation is the one that you are playing.
Hope I helped, have a nice day.