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?
Necromancy is not inherently evil
Necromancy spells are only characterized by their use of negative energy; negative energy is entropic in nature but is not a moral force. The inflict line of spells, for example, aren't evil.
But...
Necromancy does have some evil spells. Thankfully, they'll helpfully be labeled [Evil] in the spell descriptions!
Spells associated with particular alignments are indicated by the chaotic, evil, good, and lawful descriptors in their spell descriptions.
Per the Cleric class description, the Cleric cannot use spells with an alignment component opposed to their own; Good Clerics can't use [Evil] spells, Evil Clerics can't use [Good] ones. When in doubt, just check the spell's description!
Further, Channel Energy does not affect non-spontaneous spellcasting. While Channel Energy determines what spells a Cleric may cast without preparing, it doesn't affect spells that the Cleric memorizes ahead of time and does not otherwise restrict their casting list, though their alignment may do so (see above).
Best Answer
Increased characterization will provide interesting theoretical optimization challenges, reduce the strategic options of the DM, and make it more likely for you to actually play a game session.
I'm actually going to expand the question slightly:
What are the benefits of increased characterization?
As stated, any attempt at characterization theoretically creates more in-world reasons for the DM to attack you. Be it a character goal, alignment, or what have you.
We must address three things: The opportunity cost of characterization to the DM, the opportunity cost to the player, and the benefits to the player. In many ways, I will be drawing on my character optimization paper here.
What it allows the DM to do and not do
An alignment, or characterization, allows the DM to structure the world and opponents to more closely align with the character's goals. If the character is a good character desiring power, a kindly (but not too deep) DM will present evil opponents inhabiting the world to provide a guilt-free slaying experience, because that is, apparently, what good people do in D&D.
But, more to the point for an optimizer, it constrains the DM's decision space. Given that the DM can place any enchantment on a monster and that consecrate and the various $antiAlignment spells exist, a monster who takes those spells cannot take others. Given that you know the more likely set of spells taken/enchanted by monsters, it becomes trivial to then prevent those spells from providing any utility to the DM through strategically chosen counter-spells.
To summarize: characterization choices you make restrict the "logical" (in terms of story) courses of action the DM is likely to take, making it easier to foil those courses of action. Given the non-optimal nature of anti-alignment spells and effects, you can entice the DM into sub-optimal "play" through increased characterization.
Opportunity costs to the player
The act of characteriazation, be it through alignments or a more nuanced moral code, is the act of articulating and playing a series of prescriptions and proscriptions: acts that you would prefer to do and not do. Theoretically speaking, this absolutely limits the gross decision space of possible actions you may take. However, the addition of structure via pre- and pro-scriptions into your decision space allows for more nuanced choices, strategies, and options.
Structure, while nominally forbidding actions, creates a logical framework that improve the coherency of the action space as a whole, while allowing "forbidden" actions, through an act of reframing, to both be allowed and character-affirming.
Benefits to the player
As an optimizer, but as someone who enjoys RP, there is an interesting dichotomy in play. The problem I all-too-often finding myself having is weighing otherwise equal mechanical choices. By creating a narrative framework and narrative requirements, I find that certain choices by their appropriateness to the narrative, become more attractive.
Though, while looking at your bio, I found this:
The best benefit of characterization, and the best benefit of choosing alignments to support a narrative structure is: people will be more interested in playing with/DMing for you.
A theoretical character that can theoretically beat Pun-Pun a level before he hatches is all well and good, but it still lacks the necessary social prerequisites that transform it from a fascinating mathematical exercise into a game. By intentionally creating restrictions and requirements for your characters, you can present yourself with a greater optimization challenge (which I personally find more fun then bland already-solved puzzles of max damage), and allow for opportunities to bring these theoretical abstractions into play.
Edit (based on reflection):
Weakness is a poorly framed statement.
This statement, more than anything, is a framing problem(caution, SEP). As alignment is representative of characterization, and you view characterization as a threat to optimization, you can then weight your ontological values placed on the (trivially counterable magics and items) such that they prohibit you from making any other choice than the one you just made.
Asking you to update your fundamental belief system based on an answer is silly. And while I (think I) make persuasive comments above vis-a-vis characterization, if you don't want to do it, you don't have to.
Here is some logic to reduce your confidence in your belief of true neutral's superiority: