'My Word Is My Bond' and 'Brought the Law' basically rely on raising your Honor and Law values, respectively. For 'With An Iron Hand,' have a really high Order stat.
'Frontier Freedom' involves decreasing Order, I believe, and 'No Rules to Play By' involves decreasing your Law stat.
Now let's get into the less straightforward achievements... SPOILERS SPOILERS SPOILERS
The 'K-Bar' award is pretty simple. Either bring Dan Schmidt with you when you go out to investigate the river in Chapter 1, or bring Caraway and have a decent Gunfighting stat. Bam, dead bear. I recommend butchering it for the hide and meat; they can be sold or used to raise stats.
'I Am The Marshal' is also quite simple: after the climactic battle, stick around to deal with the Uptons instead of leaving the county.
'True Indian Friend': Help Yiska help the Shoshone. Fight the miners on his behalf (but don't kill them, just pillage the everloving hell out of them), be courteous to Taboonsheya and offer him potlach, and negotiate with Many Falling Rocks instead of fighting them.
The Nameless Marshal: Keep your notoriety at zero. This means obeying the letter of the law (i.e. don't murder anyone and negotiate with the Admunsens instead of plundering their camp), don't look at the dime novels on sale in Matthew's general store, and don't talk to the telegraph operator.
Ride That Bear!: Instead of killing the bear, have a really high Brawling or Intimidate stat. If Brawling, wrestle the bear, then let it go. If Intimidate, wait for the bear to look at you, then yell at it.
Later on, you need to get into the Upton camp. (You can do this by choosing to parley with them when you track down the raiders; as long as you have a low-enough value in Honor, Law or Order, they'll accept you.) The bear will be chained up in front of Reginald's tent, beaten and unhappy. After you talk to Reginald, there'll be an option to step outside and release the bear. Follow the bear as it tracks down Reginald, and you'll get your chance to ride it.
Love to Love: Seduce all five love interests: Maria, Caraway, Dan, Preston, and Yiska.
I generally start with Caraway and then seduce Dan, Yiska, and Maria around the time of the Steele trial. (Obviously, this involves doing things that raise their relationship with you; be civil to Maria and don't lynch Steele, judge the first encounter with Dan in his favor, don't hurt Caraway, and help Yiska help the Shoshone WITHOUT KILLING THE MINERS.) Preston comes later; after you've saved Preston Springs from the flood, you have the option to get mad at him, then slap him. Cue a faceful of Town Sheriff.
Don't tell any of them about the others, or try to break up with them. Late in the game, there'll be a scene where they get together and demand that you pick one. Pick one or none.
Master Betrayer: For this, you have to join up with the Uptons, as described earlier, and have already achieved Love to Love. Choose a love interest who is not either Schmidt or Caraway. Then go to the Upton camp (bringing only Carson), then kill Reginald and talk his band into joining you (using high Intimidate, high Persuasion, or by riding the bear). Take the band to loot Preston Springs. Carson will turn on you because he's a man of principle, and Schmidt and Caraway will show up to go yandere on your ass. Kill them all.
Hang the Law: First, build up an ironclad case against Marshal Steele during the impromptu 'trial.' Keep Andy Smith alive (this requires being a doctor or being struck by lightning), get helpful telegrams, get testimony from Dan and Mother Maddy. Eventually, if you've convinced the townsfolk enough and have said everything you can, there'll be an option to let them lynch Steele or rescue him. Let them lynch Steele.
Later, loot Preston Springs using the Uptons' bandits (described above). Marshall James will confront you. Shoot Julie during the conversation with Marshall James, then whack James over the head. You should get an option to hang Marshall James.
Blackest Heart: Kill everyone the first chance you get (Denny Baso, Carson, Schmidt, the miners) and then eventually loot Preston Springs. Basically, play like an ax-crazy maniac and you should do fine.