For this achievement you will have to access the secret area in the Temples by doing some tasks in each level of the game. This means you'll have to start in the Mines and make your way to just about the end of the game.
Mines (Udjat Eye)
Here you need to collect the Udjat Eye, which is located in the locked chest on one of the stages. Find the gold key and carry it to the chest to receive the Udjat Eye. If you find a chest, or you find a gold key, its counterpart will always be on that stage.
Here you need to collect the Ankh, which is located in the Black Market. To find the Black Market, follow the rapid blinking of the eye on your HUD and beeping noise coming from it. When the beeping and blinking has become very rapid, you'll be close. Sometimes the Black Market spawns deep in the terrain, and sometimes even in hard to reach areas above you. Bring lots of bombs, and if possible, paste for sticky bombs.
Once inside the Black Market, find the shop on the right side of the stage dedicated to the strange gold object. You will need to either purchase the Ankh for $50,000 or steal it. Stealing it will cause the shopkeepers to go insane, and you will have a hard time staying alive. Your best option is to purchase it. The Ankh gives you the ability to be resurrected upon death, but you cannot die until the next step!
Ice Caves (Hedjet Helmet)
Here you need to collect the Hedjet Helmet, so while you're in the Ice Caves, look for the giant statue of a face with an Ankh engraved on its forehead. Once you have found it, die anywhere on that stage and you will resurrected near the head and teleported inside it. You will receive the Hedjet Helmet immediately and you can exit to the next level from there.
Here you need to collect the Sceptre, which is the final piece to unlocking the City of Gold. To get this, you have to kill an Anubis, which is the Egyptian looking enemy shooting purple orbs at you. It can be very difficult, so use a weapon like a shotgun, freeze ray, plasma cannon, or sticky bombs. He'll take three hits from each. One you pick up his Sceptre, look around the Temple for a golden door. It will open if you have all the required items and the achievement will unlock.
Spock wishes to Credit: DROGTURIST for this solution.