Hi - This was one of the quicker achievements in Surviving Mars that I have come across so far. You need not provide the domes with water, oxygen, or power if you are just going for the achievement. Here was my approach to get this done in around 30 minutes by sol 16:
Play as India like the achievement says. I played as rocket scientist as the class to have a 4th rocket, but ended up not using it. I brought 1 drone hub, 1 RC rover, and as many machine parts and concrete as I could fit on the first rocket. Choose a landing spot with high amounts of concrete and metal. I found a spot that was 100% available to build to make sure I had room for the domes, but that was not a constraint (just makes it faster/easier). Once I landed, I sent rocket 2 with 40 polymers ( you need 8 per dome X 5 domes), another few machines parts, an RC transport to help move materials around.
Proceed to build the drone hub in a central location for maximum coverage so the drones can work. Power it with a solar panel or wind turbine. Proceed to build 1-2 concrete extractors (power them as you so choose) to get your supply of concrete going. You need 320 concrete in total for the 5 domes. You also need 80 metal, so have your drones or RC transport mine that and move it near where you will build the domes.
You can queue up all 5 domes at once, and then sit back and wait for your drones to finish their work. Just make sure that either the RC rover or the drone hub has coverage for the materials and the dome locations. If you want to further speed up the process a little, you can send rockets 3 and 4 with whatever supplies (metal or concrete) that you need.