Check out Belkar, from the Order of the Stick. A chaotic evil character in a mostly-good-aligned party. See this quote:
Despite his kill-first who-cares-about-asking-questions attitude and lack of party loyalty causing the occasional problem, Belkar has proven to be generally effective as a party member [...] but, with carefully applied threats, the rest of the group have generally been able to keep the halfling under control and they consider him to be one of their own.
Note, though, that the OOTS keeping Belkar around has always been a bit of a stretch for willing suspension of disbelief, allowed mostly for narrative purposes, and often lampshaded by the characters themselves, so it might be a bit of a problem getting the rest of your party behind the idea of sheltering a psycho killer, but if you're fun and over-the-top enough, they might appreciate your dramatic value.
Another option that some might not like is getting explicit buy-in from the players, or at least some you think will be into it. Some might see it as a form of meta-gaming that takes away from the real roleplaying, but others might see it as a social contract that allows the party to play the roles they like and build the narratives they like.
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
Don’t play chaotic evil characters like that
It seems somewhat obvious so I was hesitant even typing this answer, but this isn’t the first time I’ve seen questions of this sort crop up so I might as well.
Well, no. Don’t do that. Certainly such a character is chaotic evil according to most descriptions, but chaotic evil does not necessarily imply “unpredictable mass murderer with zero social functioning”, the same way “lawful good” does not necessarily imply that the person is completely incapable of tolerating infractions to “the law” or will never do anything that anyone might consider evil.
A chaotic evil character doesn’t even necessarily have to be a murderer at all. It’s entirely possible, and in fact likely that such a character would draw the line somewhere. Even if a character is a murderer, that doesn’t mean they’re indiscriminate. Normally people do something with some goal in mind. A chaotic evil murderer doesn’t necessarily enjoy killing, they might just see it as a viable way to get what they want - whatever that might be. In line with that, a chaotic evil character may well like the other party members. They might be the only people they’ve ever met who’ll put up with their antics, and they may well appreciate that.
Put some restraint on the character, some sort of consistent characterisation that makes them function in a party – not really that different from playing characters from other alignments in that regard – and it’ll work out much better.
TL;DR
The problems you’re describing come from playing an evil character like they’re an unhinged indiscriminate murder hobo. Not everyone who can accurately be described as “evil” is an unhinged indiscriminate murder hobo, even on the chaotic end.