Starting a new campaign does not erase achievements, unlocked portraits, or decals.
All it does is start the missions over, from the beginning, so you can, for instance, take the alternate routes on the 3 "choice" missions (and see the after-mission movies, which you aren't privy to when simply replaying the mission via the archive).
It may also be the only way to lower your elapsed time for the "Hurry Up it's Raid Night" achievement. (On account of being able to skip some missions)
In Tidalis, a chain refers to blocks removed by a single stream, and a combo refers to blocks removed by streams triggered by blocks falling after a chain, specifically how many times this happens in succession. A combo which is "6 deep" means a combo where blocks are removed 6 times based on (and including) the initial stream. (I'm not 100% certain how it treats multiple successful streams caused by the same drop, but I think it counts them seperately.)
The simplest possible 6 combo would be something like this:
***
###
***
###
***
###
where * and # represent two different colors.
To get that 6 combo, you would simply need to align all of the blocks such that you can remove one "row" at a time with a single stream. Using < and > to show arrows pointing left/right, that means you'd arrange them like this:
>>>
<<<
>>>
<<<
>>>
<<<
You can them remove the bottom row, which will cause the tiles above to fall, which will start a new stream on the bottom row, which will cause the tiles above to fall, and so on until all 6 rows are gone.
In practice, you probably won't get that exact scenario, so you'll have to think a bit ahead - start with something near the bottom, figure out how the bricks fall, and set up a combo. Remember that if you're quick enough, you can set these up on the fly, while streams are flying, so you don't necessarily have to map out the entire chain in advance.
If you're having trouble setting it up in a normal game, create a custom game with fewer colors (3-4 is a good choice) and choose Zen mode. It should be fairly easy to set up such a scenario in that case, and once you get it, click the flag to finish the level and get your achievement.
In the video you link to, the player appears to be (ab)using the fact that the game doesn't wait for combos to end when the time limit expires. Because there's only one color, it's nearly guaranteed that some stream can be generated from the falling blocks, so he just sets up one big playing field, and lets it run. When the time limit expires, a 6 combo will almost surely have been reached, and the achievement will be granted at the end of the game.
Best Answer
The requirement means:
Once I conquered all of the remaining Danelaw territory, an event appeared proclaiming that the Danelaw title was destroyed and all Danelaw claimants lose their claim, only England remains.