Start it from cold. If it rattles like a diesel, it need a new inlet cam variator. The part's not expensive, but fitting it usually involves a belt change. It won't damage the engine, but you also won't get proper performance until it's fixed. €500 at a main dealers.
Check the oil. If it's black or low, walk away - these engines need to be looked after, and some do burn a bit of oil.
Timing belt interval is every 36,000 miles. If this hasn't been done, factor it into your price - it's expensive (€400 or so).
Suspension is the biggest problem with 147s. Drive the car over speed bumps at low speed. Listen for a squeaking (wishbones) or rattle (anti-roll bar) from the front end. Wishbones are about €150~200 to fix, the ARB is nearer €400 as it's a complete ***** to swap, by all accounts. Again, those are main-dealer prices. Bushes tend to crack, and may need to be replaced.
Other than that, they're sound cars. Engines are reliable, gearboxes good, but as usual ask your mechanic about the condition of the clutch. Electrics are generally very good on Alfas, with one exception: some models had a problem with the body control computer, which can cause just about everything to fail or go a bit mad, and it's an expensive fix.
Oh, and if the car you're looking at is black, registered 02-C-13382, say hi... that was my first Alfa