Set up a new game and select "China" as your sponsor. Choose a Commender Profile you're comfortable with - I personally prefer the Inventor profile, since maintenance- and power-free Drone Hubs are a fantastic thing to rely on, but you could also benefit from the free Shuttle Hub tech to aid in distributed colonies. To make it even easier on yourself, select an easy landing site, like Elysium Beta (2N176E).
The achievement itself seems a lot more hectic than it actually ended up being. I reached 200 by Sol 55. The goal of this game is to set up a solid infrastructure, build six or seven small domes, and then keep shuttling people to Mars. You do not have to rely on births to get this achievement.
One of China's perks is that they can send 10 more applicants per rocket, so each Passenger ship can (and should) have 22 people on it. This virtue of this is you can set your first dome around a Rare Metals deposit and build multiple extractors, relying on routine exports to keep your funding up. With three total rockets, I typically had one set to auto-export, one for passengers, and the rest to bring in Machine Parts as needed. (I had a Polymer factory set up by the time of my second Dome and that helped keep costs down.)
My timeline looked like this:
Sols 1 - 15 - Saved and reloaded as needed to use Orbital Probes to find a good landing site near two concrete deposits, two rare metal deposits, and a water deposit. Landed and got a basic infrastructure going (wind/solar with batteries, five water extractors and many water towers with fuel refineries, MOXIEs, etc.)
Sol 16 - Built my first dome and brought in 22 folks. Founder stage began, which slowed things down a bit. Prioritized the following buildings in my dome: two basic homes (14 pop each), two farms, and two rare metal deposits. Grocer, diner, and infirmary for basic services. Nature stuff in the rest of the spaces. I also used this time to use my Transport to pick up all the surface metals in the nearby area.
Sol 26 - Once the founder stage was over, go ham on building domes and copying the first dome's layout. Once Food was going I went down to one farm per dome and filled that space with other production buildings. I also added a Polymer factory to my first dome, since that population was topped off.
Sol 27 - 52 - Repeat buying Cargo shipments of machine parts (the only resource I couldn't reliably make) and any other things I needed (one cargo rocket was just drones, for instance). Export rare metals to make money. Exploit technologies to make maintenance cheaper and/or less frequent. Address crises as they come up.
The nice thing about China's second perk (more applicants) is you can reliably have a full Passenger ship of 22 colonists every few Sols. This keeps your workforce up and allows you to generate a lot of exportable goods. Based on your breakthroughs, this run could be made even easier especially if your technologies aid in colonist health and wellbeing.
Edit (8 May 2018) - With the introduction of Passages, this achievement should be even easier to get, as you can link together domes now and they'll share populations and services.