In general in play they were ignored or just treated as an abstract language with no further comment.
As to where they came from, here's an answer from Gary Gygax on Dragonsfoot!
As D&D was being quantified and qualified by the publication of the supplemental rules booklets. I decided that Thieves' cant should not be the only secret language. Thus alignment languages come into play, the rational [sic] being they were akin to Hebrew for Jewish and Latin for Roman Catholic persons.
I have since regretted the addition, as the non-cleric user would have only a limited vocabulary, and little cound [sic] be conveyed or understoon [sic] by the use of an alignment language between non-clerical users.
If the DMs would have restricted the use of alignment languages--done mainly because I insisted on that as I should have--then the concept is vaible [sic]. In my view the secret societies of alignment would be pantheonic, known to the clerics of that belief system and special orders of laity only. The ordinary faithful would know only a few words, more or less for recognition.
In other words, it was supposed to be more like religious languages, but wasn't really well thought through. It disappeared in Second Edition and was not missed.
but how would performing these almost surely evil acts affect how my character is expected to play the game from then on?
To me this is the most important thing you asked. The answer is: Not At All.
Your character appears to have a well-defined backstory and personalty. Those things define how your character should act. Alignment should reflect those, and that's all. Your alignment changing because of your actions should NOT affect how your character acts, as the alignment just shifted to reflect what you were already doing.
Don't fall into the trap of going "oh I got revenge and the DM made me evil, so now I'm throwing out my character's persona and going supervillan."
Alignment is there to help people who don't have a clear personality for characters, provide game mechanics for things like Smite Evil, and provide some restrictions that we could probably do without. It's not there to act as a straightjacket on what is a well-developed character.
Now, the actual act of getting revenge and fallout from that might cause your character to behave differently. That's RP and entirely in-bounds. If the party treats you differently afterwards that is going to have an effect, but the alignment system doesn't reflect that very well. (Parties can shun good aligned people too.)
How much should other players be aware of my plans?
Depends on the players, really. Are they players who will be okay with some manipulation and secrecy? Or are they players who really value team cohesion and won't appreciate being led around for a secret potentially evil plan?
There's not a good way for me to answer that, as you know them better than I do. I would say that your character probably won't tell them more than necessary unless he's sure that telling them would make them more willing to help.
How careful should I be about players' reactions to having my PC manipulate other PCs?
Same thing as above, really. Some players would have no problem with it at all, others would be rather unhappy.
If my character does manage to enact his revenge (and he may never finish doing so), assuming the actions are metaphysically evil (e.g., knowingly killing a good-aligned entity, etc.), does this make my character evil? What about the other PCs - are they evil? If not, do they turn on me?
While killing a Good creature like an Angel could be called an outright evil act, one evil act doesn't necessarily make you evil. Mortals almost never follow alignments perfectly, they're a mix of different acts and it's more the trend that determines it.
The other PCs may or may not have their own alignment trouble, depending on the situation. How much do they know about why they're helping you? How good are they usually?
And no, if they're helping you do this knowingly and your alignment shifts because of it, they shouldn't just automatically turn on you. That wouldn't make sense, would it?
Best Answer
Just eliminate it entirely as a really poor design decision in the first place. It’s totally not necessary.
The effects of the template should be role-played. So the character should have to struggle with primal urges and the like, and the player should try to make this interesting and part of his character’s story. But his alignment should not arbitrarily just change. The alignment change should be treated as a thing that happens to the weak-willed, perhaps, or to commoners unprepared for the magical assault on their bodies, minds, and souls (as adventurers, even low-level ones, might be). It might be a thing that could happen to a player, if he indulged those urges. But not automatic.
I don’t particularly like the idea of Will saves vs. compulsion to enact said urges. A good roleplayer shouldn’t need to be forced, and ultimately the effects of a few bad rolls in a row could be very problematic for keeping the player’s character as the character he wants to play. This kind of mechanic could be done well, but ultimately I’d rather just make it a plot-device that the player can play with. And if the player isn’t interested in doing so, then I really don’t think making him roll Will saves is a good idea: it is a game, after all. Forcing the player to “play” a game he doesn’t enjoy doesn’t make much sense, and strikes me as rude.