The way those missions typically work, unless the target is only listed as being found in a single system from the start:
- Once you're in the target systems, you should get a 'tip' encounter within one of the first few USS that pops, for me it often ends up being the very first. It seems to be even more common if you've already been through every recommended target system.
- The 'tip' will be a pilot telling you exactly what system the target was last spotted in. it may not be one of the initial recommended systems, but it'll still be nearby to where you got the mission.
- The target will always be in this system from then on, and MUST be killed within this system. If they flee (normally they won't unless you summon the cops, sometimes not even then), don't bother scanning the wake because chasing them into a new system won't count towards the quest.
- The target will almost always be one of the first USS you encounter in the 'tipped' target system, unless you have a bunch of other quests queued up that also offer tips or updates, since those can take priority.
- If your target is named Gen. Something (these might only happen in warzones?), they're significantly more likely to be an elite Federal Dropship or Clipper than an elite Anaconda. This does not necessarily make the fight easier, depending on their weapon loadout, but it does at least mean their shields will go down quicker. Generals also usually aren't Wanted, so expect to acquire a bounty if you do these.
- Probably a good idea to turn off Report Crimes Against Me in the options of the right cockpit panel. Some criminals will start charging FSD as soon as the feds arrive, especially in rare cases where the authority decides to appear on the scene with an Anaconda or Python.
- There's a bug that causes ALL assassinations of the same type within the same system to be completed with a single kill, even if it's a different person you're targeting. To do this, all of those quests must already have been 'tipped' to be able to complete in this way. I discovered it by accident and earned about 1.2mil all at once, and have been mainly trying to avoid it since then for the sake of not exploiting the game. If you do want to use this method, pay attention to the organization that the target is associated with (if any) when accepting the mission, as this will often tell you where their final location is likely to be (for example, if your target is a pirate lord associated with the Freng Organization, it's practically guaranteed that you'll need to eventually kill them in Freng).
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As for tactics and time taken, it really depends on your ship. If searching for the target is the only thing on my table, then I can usually hit the correct set of USS within 5 or 10 minutes, probably less especially if I've got multiple active Assassinations so I can get multiple tips within the same system. The fight itself could take 20 minutes with something like an Eagle, or it could be over before the police arrive if you're in something like a combat-specced Asp or other larger ship. I find Beams (or your preferred laser type) to generally be the best option against Anacondas, as they bring shields down more quickly while still doing respectable damage to sub-systems.
In general, the best tip I can give is to familiarize yourself with using thrusters rather than standard AI-style fighting of spinning in circles or jousting, as either of those strategies are going to get you damaged heavily and quickly. Use up/down or left/right thrust combined with backwards thrust or reverse throttle to orbit the target at a certain distance. Also make sure that you are targeting either their Drive (so they can't run away or control themselves and become a sitting duck, it's also a larger and weaker target) or Power Plant (reducing this to 0 instantly destroys them).
If you're not confident in fighting them yet, generally try to stay right on the cusp of 2.5 or 3km while orbiting them, just within weapons range but too far out for their idiot pilots to hit you with anything but inaccurate gimbals so long as you keep moving. If you manage to get at their rear/underside area and feel like you're agile enough to stay there, especially for the anacondas, try to stick close by and probably within 1km. That way they'll have trouble turning to track you and the only threat to your well-being will be whatever turrets they happen to have on that side of the ship. Speaking of, pay attention to their turret loadout based on what you see hitting you - if it's burst or pulse lasers, you're mostly very safe sticking towards that side of the ship, but if it's a Beam Laser then you may want to try your luck on the other side of them or be forced to burn Shield Cells. Ideally they will have a weak turret (or no turret) on the ship's underbelly, as this is also the best location to deal damage to the Power Plant on an anaconda or dropship.
If they do knock out your shields, leave the 3km range as quickly as possible unless you're really confident in your hull armor. At close range even mediocre turrets will eat into your hull, and at long range you're gonna get blasted mainly in the cockpit and risk losing your windscreen surprisingly quickly.
And this probably goes without saying, but Never Ever Ever get directly in front of the anaconda - that's where they keep the Plasma Accelerator that can potentially one-shot anything lighter than a viper or cobra, even with shields up. The first time I tried an assassination, I took a hit from one at full health in my Viper and managed to limp away into FSD with 1% hull remaining.
Good Hunting