These days £5k is an excellent budget for a 530i Sport and will allow you to pick from all but the overpriced ultra-mint examples from optimistic sellers.
The pick of the range as standard is the 530i Champagne II edition. 144 of these were registered in the UK as a runout special edition in 2003. It's a standard 530i Sport with Navigation but comes in the unique Individual colour combination of black paintwork with champagne coloured leather and a unique colour wood trim. You also get the rare extended leather option as standard on these. There is also an identical Aegean Blue Edition which is pretty much the same thing but is in Aagean Blue with grey/white leather instead. The interior on this isn't anything like as awesome as the Champagne II, though. Both editions are automatic only and have the best E39 wheels - 18" Style 37's - as standard.
If you want a manual you are in for a search - only 431 were sold originally and there were no special editions in manual form. For a number of years from new they were worth less than autos but now they tend to fetch more and are in more demand than the autos, I guess because the ownership demographic of a £5k E39 has rather different tastes to that of a £20k-£30k car.
I'd prioritise condition, history and number of owners over specification at this point in the E39's life. The standard spec on a 530i Sport is pretty excellent anyway - DSC, Cruise Control, Dual Zone Climate control with auto recirculation, etc etc are all standard fit and most of the options are either simply 'nice to have', out of date or not particularly challenging to retrofit these days. The only exception to this is leather - you don't want one without leather.
There are a plethora of common issues - the two biggest being autobox failure and cooling system problems. The autobox issue is probably the only 4 figure bill left on an E39, the cooling system can be renewed for £500. Most E39's are also due a suspension refresh - bushes and dampers get tired and increasingly less fussy owners tend not to notice or do anything about it. Less than a grand to replace all the suspenson arms, springs and dampers though.
The standard wheels on the 530i Sport are the 17" Staggered Style 66's. They are 235/45/17 front and 255/40/17 rear. Premium tyres from Continental or Goodyear will cost you £120ish each for the fronts and £160ish each for the rears. The optional wheel on the 530i Sport was the 18" staggered Style 37. This runs 235/40/18 front and 265/34/18 rear. Premium tyres will cost you £150ish each on the front and £210ish each on the rear.
I have owned mine for 6 years this year, and in that time it has cost me on average £400 a year on tyres and about £800 a year in fixing broken stuff, preventative maintenance and routine servicing. I think this is a pretty useful average (I managed to go a whole year once spending nothing other than insurance tax and fuel yet some months it's really made me wonder what the point is) to work with. Keep a grand spare for a car fund and top it up as needed and you'll be fine. Lots of the things which used to cost enormous amounts of money to fix can now be done properly for less - the DSC units sometimes fail, this was once a £1700 bill for a new hydro unit, you can now get it refurbished for about £300-400 for instance. Other bills remain as much as they were when the car was a £30k car if not more so, for example tyres.
Servicing in the main dealer network is reasonable - they offer value servicing rates and you can often haggle. My last two services for instance have been an Oil Service at £120, an Inspection II (The major one) at £330. You can get it done cheaper at an indy but I've never found the savings worth the bother for routine work.
They are fantastic cars. I love mine and I never in a million years expected to still have it after this long. It's just brilliant and I love driving it, often inventing sometimes ridiculous trips just to give me a chance to drive it some more. Doing 1400 miles up to and around Scotland in a months time for no real reason other than it'll be awesome.
Here are a few pictures of mine:
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I look after my cars but the bodywork on yours is outstanding!