How can I make Strong like me without compromising myself?
The wiki says Strong (and I have noticed that Curie and Hancock as well) likes to be out and kill things... And that gains approval even if it does not seem that way. Allow me to elaborate; you can get strong to adore/idolize you simply by going out, exploring and killing. So you do not necessarily need to do things your companion "loves" or "likes" for him to gain approval. I can confirm this is true for Curie and many people seem to agree it works for Strong too.
So my suggestion? Go to Diamond City, check the bounty board (that's left to the guy selling "swatters") and do a lot of them with him. Most bountys just requiere you to go somewhere, kill the boss of the place and loot. So do that, but remember never to lockpick.
What I am trying to say, is that the following is without logic, but still happens: I left Codsworth at Sanctuary, and you notice that he "loves" when you customize a weapon or an armor, and so I intentionally only customize things at Sanctuary to have him "loving" it, but how can it be that it took me 8 "loves" just to reach the first level of approval, and with Curie i have barely got one love and 4 likes and she is already in 3rd level of approval with my character? Well it seems that "hanging out" and "watching each other's back" raise approval silently. So you most definetelly want to take strong with you to some raider camps and gunner strongholds. He will "approve" just being there as a team teamwork. I hope someone with PC and console commands may explain how does this works (because it does) in terms of how much time spent, or xp gained, or enemies killed it takes to get "likes" equivalent approval... But that's that.
If you have trouble finding the bounty board, go to the mayor's office and ask the secretary for work. The bounty board will appear on your map after that.
While Pedro's answer is fine, and will certainly work given time, I found a few strong shortcuts.
The biggest one is the quest Confidence Man from the Diamond City bar. The quest provides you with a number of dialog opportunities that will make Strong very happy, as you encourage Travis to be less of a weakling and to fight more.
Similarly, the quest The Big Dig in Goodneighbor, offers a number of intimidation opportunities that Strong is quite pleased by.
The Secret of Cabot House quest chain also offers a large variety of Strong friendly dialog, along with lots of tasty raiders to shoot up.
Finally, Diamond City Blues is a cornucopia of violent and threatening dialog, and the mission itself, being a string of chem deals, is right up Strong's alley. He'll love you for it.
Also, if you're going the Radiant quest route, picking up quests from Settlements around the commonwealth (Hostage rescues, clearing out raiders, etc.; Minutemen stuff), by agreeing to take the quests immediately without asking for more caps, Strong will approve of your generosity, as well as of the subsequent killing, which is a great way to double dip.
Strong likes it when you help settlements for the Minutemen.