Digging It Up
I would suggest working a few level one chests to get acquainted with everything before going to the tougher chests. When you dig up your chest, you will also dig up guardian spawn. Levels 2 and higher will spawn 4 guardians when the chest is completely dug up.
Once you start digging, as long as you are in the correct location, you will see the ground start changing to dirt and see the chest start to rise out of the ground. You need to precast invisibility but do not click on yourself to apply the spell until you see the entire chest pop up. There is a very, very short delay from when you see the entire chest appear to when the guardians spawn. If you have timed it right, they will be standing around you and start to wander. If you did not time it right then you are either running for your life or you were revealed and are now running for your life.
Once you have killed the guardian spawn, you need to open the chest. Use your lock picks on the chest. Once the lock has been picked, you need to be very careful as the chest has been trapped. You can use the skills detect hidden and remove trap or you can move 7 tiles away from the chest and cast telekinesis to set off the trap. After it has been removed or disarmed, it is safe to open the chest.
Now you are ready to loot the chest but be careful as you do, guardians will randomly spawn as you remove items from the chest. What I like to do is “work the chest” which means leaving one item, I usually leave a single scroll, and then you drag it a little ways from the chest and let it go, it bounces back in the chest and you can spawn some extra guardians for fun and extra loot.
Added by Basara:
An alternative to precasting invisibility, especially for melee-based Treasure Hunters (also, Mages for level 6s), is to have a rune marked nearby (several screens away - substitute the key of a ship, for small islands), and have Sacred Journey (or Recall) precast, targeting the rune/key when the chest comes up. This allows a Melee Treasure Hunter (or any of the other templates that need to isolate the spawn, and take them down one at a time), to get distance from the spawn, to where they all are not automatically trying to attack you.
Another, minor, advantage it has over invisibility is that it removes you from the chest area so that mis-timing the apply of the spell (or being instantly revealed/detected) doesn't get you pummeled into the ground before you can react to escape. If the spawn still chases you from the chest location, despite the recall away, you then have more time to successfully recast invisibility, hide, or even use an invisibility potion (which has a timer before it kicks in, and therefore isn't all that effective to be used while digging).
The down side to this is there are several islands that have server lines through them, and you do not want to recall across a line and incur the "speedhacking monster" bug that can happen when a monster doesn't lose lock on you from a recall, then gets abnormally sped up when it chases you across the server line. The islands of Iver's Rounding, Buccaneer's Den and Eiffel Island are notoriously bad for this.