Some tips for scoring in Tedi Hardcore:
1) Start off with one mineral digger, who you can direct towards different mineral patches that are not attached to caves.
2) Seals are expensive, so only seal caves close by, but always try and instant seal the entrance, so that lava won't keep moving about. It is also more efficient to dynamite dwarfs upon discovering minions, as warriors are expensive too.
3) Only build an outpost if a boss spawns, until you're well into your run. Usually you'll only end up unveiling caves with shamans in them. I'd recommend trainign 4 knights to level 3-4 and launching them all at once to defeat it. Again, instant dynamite on the dwarf who discovers it will help a lot as it eliminates extra goblins.
4) Dynamite is cheap in comparison to a mistake causing water/lava spreading. I recommended using dynamite on any dwarf that strays too far from your revealed radius. This is to make it easier to keep track of when your dwarfs discover caves, so you can deal with them accordingly. Keep the one mineral miner alive though, but keep an eye on him.
5) Once you get rolling, you can start up another mineral miner in another direction if you can multitask, as you'll need a lot of money for seals and outposts on Tedi Hardcore.
I've included a video of myself getting over 1 million score on Tedi Hardcore, which hopefully illustrates my advice and allows you to score at least 500k for this achievement. Godspeed!