Use thrusters to maneuver and to orbit it and to stay at your preferred distance. If you try to fight it like every other crappy AI pirate by going into circle-of-death turning battles, or even worse by jousting at it with boosts, you're going to get killed since the Elite level AI generally knows how to use FA-off turning.
The thing is, even as Elites, their aim still isn't good enough to hit you consistently at 2.5-3km range except with inaccurate Gimbals, so if you can orbit them with thrust and stay out around max weapon range, you should be able to whittle them down while taking negligible damage. If you do happen to out-turn them and find yourself facing their underside, try to do the same orbiting but stay within 1km, where they can't quite turn quickly enough to keep up and you have free shots on their Power Plant underbelly.
It's also a good idea to equip Shield Cells if you aren't already using them, they're insanely powerful in all types of combat. They aren't always necessary when you fly well, but sometimes those assassin targets come equipped with a Beam Laser in a turret or other inconvenient position, which can eat your shields even despite its inaccuracy.
i use the the thrusters a lot in combat, but in this situation they didn't really help. i know that jousting and death circles are not the way to go - my tactic was to get very close and to its rear quickly, so i had better "access" to the blind spot on the underside. but, as said, the dropship was very quick and fast (especially considering its size), i never could stay there for long.
i will try to keep my distance next time, might actually help. i also considered a shield bank, but have not found one in the system i was in (i could have just jumped to the next system, of course....

)
I also bought myself a torpedo which I threw at him after I'd taken his shields down which took out about 25% of his hull in one go.
the sledgehammer tactic

i might try this one time.
Hello mate,
1) don't accept 150+K bounties unless you know what you're doing.
2) Get a real fighting ship, not an explorer ! Seriously... I'd recommend a Viper with 4 gimbals (word on the street is 2 small pulse/burst laser and 2 medium multicannon, but I myself fly a small burst medium pulse laser setup) or fixed if your aim is top notch.
Before engaging your target, make sure to have a subsystem targeted (FSD and power supply are vital organs, but I usually go for thrusters) and line up on their 6 (behind them, for backstabbing. It also helps seeing the thrusters in my case), most AI ship let you maneuver however you want and wait for you to open fire before turning hostile (unless they're pirates and/or hate you). Might want to grab a Kill Warrant Scanner too, while you're at it. What I usually do is set course at 5km and target subsystem, scan at 2.5km and open fire <1km.
Also learn some dogfighting tips and tricks, like Flight-Assist-Off-turreting or backward engines + vertical thrust + pitch + yaw to maximize the time your target stays in your aiming reticle.
1) i know what i am doing... at last i thought so until i got scattered throughout the eravarenth system

2) yes, i'm thinking about getting a viper for bounty hunting and make the cobra a proper explorer or small hauler. i usually use fixed weapons, i don't have much trouble aming and found the gimbals often "sweeping" over the target in rough dogfights. the dropship always faced me as soon as i dropped out of FSD, followed me when circling it and turned hostile after a few seconds. subsystem targeting is a good idea, i always forget about this

flying backwards is something i haven't done before, i will try it next time.
I've never had trouble with dropships in these missions. I usually get elite Anacondas. The dropships I've encountered pretty much always have MCs. If possible get the best shield generator, thrusters and a military bulkhead. But it is possible without these using shield cell banks and chaff. I personally wouldn't go with 4 pips to engines, but 3 eng and 3 to weapons.
i constantly change the power distribution to fit the situation. military bulkhead and upper class internals are outside my financial capabilties at the moment. teh dropships i encountered was fully equipped with pulse lasers.
The front end of a dropship is the dangerous end (obvious duh)
It has more armor than a conda, but handles like a turd and turns like a brick.
Get some shield cells, some armor piercing weapons (multicannons) and some armor upgrades (military grade composite if you have the funds.)
Avoid the front at all costs. Use your latheral thrusters to stay behind it...and avoid the front.
i learned that at my first encounter - unfortunately, the top turrets still pack quite some punch. are multicannons better against hull than cannons?
my loadout (i exchanged the cannons for fragment cannons after my fist post):

