Feats of Strength show up in other Blizzard Battle.net 2.0 games. Both Starcraft 2 and World of Warcraft Feature them.
Blizzard achievements are worth a certain number of points, which is displayed on your relevant battle.net profile, but Feats of Strength always are worth 0 points.
Feats of Strength are also not listed anywhere on the achievement list - it's not that they're hidden, but rather that the majority of them are one-time or limited-time events, and it is impossible to achieve them. (For instance, "Log on during WoW's 7th anniversary").
What this means, is that Blizzard has a way to acknowledge or commemorate players at certain times and events without affecting the number of earnable achievement points.
But don't take my word for it, this is straight from Blizzard CM Bashiok:
The distinction between a Feat of Strength and an Achievement is that
an Achievement is something you can work to achieve, and a Feat of
Strength is either limited due to an outside factor, or is just simply
impossible to get any longer.
Things like attending BlizzCon, where only a few thousand people can
attend, is a Feat of Strength.
Of course there are plenty of people who want each of the Feats of
Strength, but the reason why they aren't Achievements, and don't
provide Achievement points, is due to the limited-time, invite only,
or simply impossible nature of obtaining them.
Original Source
Contrary to the current accepted answer, it appears that you do not need to listen to dialogue for a minimum of 8 seconds. I tested this today and made a quick video of my results, see it here.
In summary, you can quickly skip through all of the dialogue and still get credit toward the achievements.
Edit: I have run more tests, here is a link to a Youtube video where I skip through 18 conversations of varying lengths at full speed and get credit for all of them. It's not exactly a riveting video, but I think that it sufficiently supports my answer. Here is the video (the footage is sped up to make the video shorter).
Best Answer
Similar to The Hero's New Clothes achievement, you literally have to unequip your weapon(s) and attack Diablo with your fists. You'll need to remove your Primary attack ability from the A button and have that slot empty.