This will be your first trophy. Simply start the very first level, collect the golden key and get to the level exit.
See this post for videos on completing and gold ranking all levels.

In every level you will find three different kinds of treasures: silver coins, gold coins and sapphires. The amount of treasures collected will affect your level rating (Bronze-Silver-Gold). You need to collect every available treasure in a level to get a gold rating in it. Do this for 5 different levels to get the trophy.
See this post for videos on completing and gold ranking all levels.

The game consists of four chapters. Complete every story-level in the first chapter (Amazon Jungle) to get this trophy.  See this post for videos on completing and gold ranking all levels.

The game consists of four chapters. Complete every story-level in the second chapter (Himalaya) to get this trophy.  See this post for videos on completing and gold ranking all levels.

The game consists of four chapters. Complete every story-level in the third chapter (Egypt) to get this trophy.
See this post for videos on completing and gold ranking all levels.

The game consists of four chapters. Complete every story-level in the fourth chapter (Maralidia) to get this trophy.  See this post for videos on completing and gold ranking all levels.

Complete every story-level in the first chapter (Amazon Jungle) with a gold rating by collecting all available treasures.  See this post for videos on completing and gold ranking all levels.

Complete every story-level in the second chapter (Himalaya) with a gold rating by collecting all available treasures.  See this post for videos on completing and gold ranking all levels.

Complete every story-level in the third chapter (Egypt) with a gold rating by collecting all available treasures.
See this post for videos on completing and gold ranking all levels.

Complete every story-level in the fourth chapter (Maralidia) with a gold rating by collecting all available treasures.  See this post for videos on completing and gold ranking all levels.

Complete every story-level in all of the four chapters with a gold rating by collecting all available treasures.
See this post for videos on completing and gold ranking all levels.

The treasures you can collect on any level have different values. A silver coin is worth 50 points, a gold coin is worth 250 points and a sapphire is worth 1000 points. To get this trophy you have to collect a total value of 250,000 points, which means that you need nearly all treasures in the game.
See this post for videos on completing and gold ranking all levels.

The score you get after finishing a level is calculated by the amount of collected treasure points per minute.This means you should collect the maximum amount of treasures in the minimum amount of time for high scores.
Sadly you can't score 6,000 points on the level Tomb of Osiris (chapter 3 level 9) while playing alone. You NEED to play in coop mode where player 1 controls Hamilton and player 2 controls the bird Sasha. The neat trick here is to let the bird fly near Hamilton all the time. While doing this, all collected treasures will count double (Hamilton collects the treasure and scores the points, the treasure flings in a little animation into the air and the bird catches it and also scores the points). You can also do it alone with two controllers. Just make sure to rapidly hit the -button on the second controller everytime you collect treasures so that the bird always appears near you.
Watch the following video for the quickest route through the level and for a better understanding of how the explained method for 6,000 points works.

While playing through the story at some point you will be introduced to the "Agents" (frog-like coat-wearing beings). The level you see your first agent you get this trophy automatically.

See the next trophy "King of Bling".

As explained before (trophy "Egyptologist") you can earn much higher scores while playing coop-mode. The easiest level to obtain this trophy is chapter 4 level 13 (bonus-level). To unlock the level you have to earn a set amount of treasure value first.
Following the same method as in trophy "Egyptologist" (leaving the bird at Hamilton's side in coop-mode) you can easily get 15,000+ points.
If you want to do it in solo-mode it is possible in chapter 2 level 15 to score 10,000 points however it is very difficult. (Thanks to p3re1ra for the tip)

To unlock an optional level you have to earn a set amount of treasure value first. But they unlock very fast one after another if you play through the story and don't leave treasures behind. Finish any of these optional levels to get the trophy.

As in the story-levels you can collect treasures and receive a rating in the optional levels too. Collect all available treasures to get gold ratings in all optional levels to get the trophy.

The sentinels will be introduced to you very early in the game. They look a little bit like golems and follow a straight path in a level. You can defeat them with different methods. For example letting a wheel roll over them or crushing them by rising a door with a switch when they run over it. You have to defeat some sentinels during the story-line to proceed so you will get this trophy automatically.

As with the sentinels the agents can be defeated too. They don't follow a fixed path but rather try to hunt Hamilton down. Agents also die when wheels roll over them, doors close beneath their feet etc. Furthermore you can distract them with the bird and moving them to a desired field by hitting the -button.

In the last level of the first chapter you meet your first Exocet (flying fishes with propellers). They won't harm Hamilton but pursuit his bird whenever you are controlling it and paralizing it for a brief amount of time. If you hit the -button when an Exocet is near you scare it and it will leave you alone for a while. The first time you do this you will get this trophy.