How does Dust work?

You have very little control over it, it's mostly the RNG.

You can get it from...

• Opening doors
  Every time you open a door there's a chance of dust dropping, which is a random amount, I think I've seen numbers from 2 to 9. The hero Max has a skill which raises the amount gained if he is the one to open the door when dust is found in a room, he gets this at level 5 according to the wiki, and it's also available on the "third eye" item.
• Kill Monsters
  I believe every time you kill a monsters there's a chance they'll drop one dust, however it might be more complicated than that. The heros Golgy and Sara have a skill which increases the likely hood of dust dropping from monsters, it looks like it only applies to monsters they kill, they get it at level 3 and 7 according to the wiki.
• Selling items
  Some merchants use dust as currency to buy and sell. If you have items you don't need, and don't expect to need, you could sell them for dust.
• Modules
  There's a Major Module called the Shop which allows you to store a Merchant, which can build Dust if it's operated while there's a Merchant in it.
• Events
  There's at least one event/tower thing where you can pay industry for the possibility of dust.

Note that dust is reset after each level, as are certain excess items, while industry is persistent, (but more dust might let you get more industry, or food, or science) which effects whether or when selling items or spending industry is worth it.

As I said, I haven't played the release version much yet, there's likely to be more specific items and modules, but that's everything I'm aware of.

Regarding tactics/strategy, it's more about adapting your controlled area/which doors you open/build things on to make use of the dust that you have, as opposed to taking specific actions that grant you more dust, since often those actions simply aren't available.

In addition to the above answer I also recently noticed on the Wiki that extra, unused Dust helps shield your crystal from attacks. (I feel this falls into "How does dust work" right?)

Source: The Crystal

The Uptake is defined as the amount of dust the player is using to power rooms. The game requires the player to uptake ten dust per room. Any dust that is not being used in Uptake is automatically used by the Crystal to shield itself. Thus, the more surplus dust you have, the more resistant it will be to damage. If the Crystal does take damage, the crystal will lose dust. If enough dust is lost, a random room will un-power. If the game cannot find any rooms to un-power, the game is over.