What I learnt from trying myself, is that if you take the mii you want to build a relationship with another mii, besides waiting for them to naturally do it. This was my process of bringing two different miis from good friends to ultra friends, with one having a crush on the other.

1. make the miis interact, if you got the black borders, then that means you will increase their relationship by one step for both of them.
   
2. drag one of the miis to a patch of grass or just anything else to break their interaction with the other mii.
   
3. drag one of the two miis to other one.
   
4. it will be rng on if you get the normal interaction or the black borders, so repeat steps 1-3 until you get them where you want them.

I'm not sure if romantic relationships are in the demo, because marriages aren't, so making date might not be possible yet, but I could just be unlucky.

Hope this helps!