So to start off, this achievement takes a fair bit of time and there is a small level or variation involved with the dates. I was unlucky and didn't have my 2nd child until Winter of year 3 but it is possible to unlock sooner. With that out of the way let's begin...
The first part of the achievement is getting married. To marry someone you must become friends with a bachelor (Alex, Elliot, Harvey, Sam, Shane or Sebastian) or a bachelorette (Abigail, Emily, Leah, Maru, Haley or Penny). You can have a same-sex marriage and adopt children. To become friends with them talk to them every day and give them a gift that they like or love twice a week; extra points if it's their birthday. A full list of gifts can be found here https://stardewvalleywiki.com/List_of_All_Gifts
. Once a character reaches 8 hearts (which can be seen in one of the menus) you have the option to buy a Bouquet from Pierre for 200g. Giving this gift to your bachelor/ette will change unlock 2 extra hearts and make them your boyfriend or girlfriend. Getting this far you should have received:
Next step is proposing which requires a few more steps. First you need to gain the other 2 hearts maxing out at 10 hearts for your boyfriend/girlfriend. Then you need to upgrade your house once to get the bedroom extension, this costs 10,000g and 450 wood from Robin. Finally you need to repair the bridge on the beach for 300g. On rainy days if you visit the island over the bridge you will see a ghost and when the previous requirements have been met he will sell you a Mermaid Pendant for 5,000g. Gifting this to your boyfriend/girlfriend is a marriage proposal. In the process you should also net:
The marriage will take place 3 days after you have proposed unless another event is happening such as a fair. After the wedding they will move into the farmhouse with you and have 2 extra hearts for a total of 12. You have to keep 10 or more hearts at all times to have a child and unlike other townspeople your spouse's friendship level will drop if not talked to. The other requirement to having a child is to upgrade your farm house again (50,000g and 150 hard wood) which will grant you this achievement:
Now you're married and have a large enough home you can finally have children. After you've been married for 7 days there is a 5% chance each night (before the game saves) your partner will ask you if you want children and even if you say no there is still a 5% chance they will ask every following night. This is where the variability is, I waited 2 seasons before my spouse finally asked. When you agree there will be a 14 day wait period before one of you gives birth or adopts. Once your child is born it will stay in the crib for 14 days, then crawl around on the floor for 28 days and finally it will be running about the place sleeping in its own bed at night. When it is sleeping in the bed you will be given a 5% chance every night of your spouse asking for another child. The sex of your second child is always different to that of your first born. When the second child is born your achievement finally pops.
So you're looking at around 1 year to propose to your spouse, 3 days to marry them, 7 days for them to settle in, average of 20 days to ask for a child, 14 days for the baby to arrive, 42 days for your first child to grow up, 20 days to ask for a 2nd child and 14 days for the 2nd baby to arrive. That's over 2 in game years minimum to make this work, longer if you're spouse doesn't ask for a child for quite some time or if you you don't keep some money back for house upgrades. At the end of the day this is a fairly chilled out game so you'll find yourself aiming for other achievements and getting this in the background, hopefully you won't have to grind it out at the end like I did. Happy farming people!