A £6k 650i is going to be a massive gamble. I ran one for 5 years ish and it treated me well, but I didn't do that many miles. They WILL throw big bills.
Valve steam seals - it's not if they fail. It's when. They all go bad eventually. If you find a 650i for under £6k that's had them done recently I'll be amazed. All the cheaper ones will be needing them doing. It's a £3k job at an independent or you can hire the tool and do it yourself. It's an arse of a job and will take 24 hours+.
Coolant transfer pipe - Another when item. Requires a complete engine strip to fix but there are stents that can be fitted for around £1k
Suspension - All 650is of this age will be on tired suspension.
If you do fancy the risk, check for the following:
Run the car up to temperature and then let it idle for 2 minutes. Rev it to 3k and watch for smoke. If the seals need doing it will chuck out a huge cloud.
Check for coolant leaks at the front of the engine. A drip is the transfer pipe issue
Check the idle speed. If it's above 600rpm then the VANOS has been disabled, usually by unplugging one of the solenoids. This is to mask a VANOS failure. The car will drive but will be gutless at low RPM and will drink fuel. Also unlikely to pass an emissions test like that.
Check for steering judder under braking. It's nearly always bad lower control arm bushings. Factor on £500 to sort that.
Check the transmission has had its fluid and filter changed at 60k ish. BMW state that the gearbox is "sealed for life" but this is only because BMW define the life of the gearbox as 100k miles. ZF (who make it) recommend a fluid and filter change at 60k. Look at £600 for that job.
Check the transmission shifts smoothly. Any jolts and it's the mechatronics unit that's at fault. £1.5k for that, but I'd walk away as it will have been putting a huge amount of stress on the clutches. It's 4th to 5th where it's most likely to jolt.
Drive it forwards in drive, then stop and put it in reverse. If it takes more than a second to engage reverse then the transmission pump is going bad or the mechatronics is bad. It will fail eventually.
Mis-fires. Can be caused by bad coil packs or bad plugs. The main reason for the plugs going bad is people not changing them. Plugs 4 and 5 are a nightmare to get to as they are right up at the back of the engine bay. Much disassembly is required to get to them.
Listen for whooshing sounds when revving. Usually a vacuum leak which can be around the rocker covers, in the PCVs, on any of the pipes leading to the PCVs, in the intake manifold, etc. etc.
Oh, just noticed you are looking at a convertible, that bring some more issues to watch for.
"Soft top not locked" warning. This is when one or multiple of the sensors in the roof go bad. Can be fixed temporarily by smacking the roof hard with your hand towards the rear, but it will come again and will need the whole roof stripping down to fix. The thing is, it will fail at the most inconvenient time. If it fails while closed, you can't open the boot. If it fails while open, well, you're getting wet. There is a manual override but it takes about 10 minutes to fully open or close the roof using that method.
Drains. The cabriolet has tons of water drains around the roof and boot area. They like to collect leaves and get blocked. When they do that they dump all the water in to the boot area where the battery and all the electrics are.
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