There's a couple of ways you can do it really. The AA will give you same-day feedback and a report within 24 hours for around £130 if memory serves. Personally I'd consider a visit to an Alfa specialist such as Autolusso, AlfaWorkshop or Alfatune as they work on these cars everyday and have a thorough knowledge of the cars, if the owner is willing to let you. As above, FSH is a must.
If not, the GT is based on the 147/156 underpinnings and has some of the same maladies.
1. Petrol engines should have their timing belts changed every 36k or 3 year (ta rockhopper)
2. Diesels every 50-60k or 4 years, change the water pump too as most timing belt failures on the diesels are caused by seized water pumps
3. Clutches will bite high if one their way out, diesels may require a dual mass flywheel with the clutch and £900 is not unreasonable for this repair. Not sure of clutch costs on the petrol engines.
4. The body panels are quite thin which means the cars pick up car park dings easily, check for overspray especially on the wheel arches which can signal a previous accident. The newer Alfa's don't rust as badly as the 'Sud's back in the day but it won't hurt to check the underside and the floorpans. Most Alfa specialists will offer undercoating to help prevent rust at around £80 a throw. Great if the roads are salty in your area.
5. Suspension - The double wishbone setup at the front is known to squeak and groan and this normally signals dry/worn upper wishbones - £100 a side including labour. The lowers should last longer (I'm still on the originals at 72k) and the rear radius arms (4 in total of unequal length) should last a while too. It's difficult to say how long they will last but if you're looking at a car with higher mileage I'd have them looked at.
6. Electrics should be sound but check everything works and GT windows are known to have a mind of their own.
This guide is a good place to start also: -
Alfa Romeo GT Buyer's Guide
Best of luck!
147 Sport Q2: Autolusso Remap + Swirl Flap & EGR Delete, Mocal Oil Cooler Kit, JTD Performance FMIC Kit + High Flow Elbow, Wizard Downpipe, PF Pads + 5.1 Fluid, Autolusso Braided Brake Lines, MTech Short-Shifter, Autolusso Silicone Turbo-to-MAF Hose, Refurbed Q2's + Some to add later.