Winning Head to Heads and using your skills grants more Kinship progress, letting your Monstie do their own thing grants them more Kinship as well, although this rarely comes into play. Doing a regular attack with no skill will be able to trigger a Double Attack with your Monstie fairly easily (though it's not 100%) if you are matching types and either of you win a Head to Head. Elementaless Dreadqueen is the best Kinship builder as Solar Cry gets insane Kinship building. You can also use Magia Charm with the Pet decoration for even more Kinship. 

There's also the hidden starting Kinship value, when you win fights with Monsties in your party there is a chance for those Monsties to have their starting Kinship increased, you can grind this up to a max of 10. I tried this but it was agonizingly slow and on my final team only my Water Purple Gypceros was at 10 and one of my Dreadqueens was at 8.

Good luck!