Tomb Raider I-III Remastered Starring Lara Croft

This is done on the level Sanctuary of the Scion. When climbing up the sphinx on the back side, do a running jump and grab the platform on the back of its head. Pull up and hold the action button at the same time to do a handstand.
The achievement should unlock.

This is done on Level 12:- Sanctuary of the Scion on the first Tomb Raider Game.

• Head up the stairs at the start of the level and defeat the 2 enemies that come down if you wish.

• Go into the room at the top, and go around the back of the block in front of you to be able to climb on top of it.

• Climb up again to get onto the sphinx's back.

• Look 45 degrees to the left to see a sloped platform that you can stand on. Climb onto that, then run, jump and grab onto the back of the sphinx head. Pull up whilst holding the WALK button ( by default on tank controls or on modern) to do a handstand.
  The TR Pet This Cat achievement in Tomb Raider I-III Remastered Starring Lara Croft worth 14 pointsDo a handstand on the Sphinxes face

If you're not getting this through the course of your playthrough and want to get to Level 12 quickly, you can use the level skip cheat 11 times to get there quickly from the start of the game. It can ONLY be done on tank controls. The level skip cheat is:-
Hold the walk button, ( by default) and push forward once to take a step, then backwards once to take a step. Release the walk button. Press and hold either left or right until Lara completes 3 full rotations. (go a little beyond your starting point to be sure). Do a standing forward jump.