HSF means heatsink-fan? Also, is that ear piercing noise going to be there in general or as a result of overclocking?
Yes, that's what HSF means. The ear piercing noise he is referring to is the noise of the fan when it is running at its highest setting (or near highest) in an attempt to cool down a much hotter processor than it is supposed to cool, thanks to an overclock.
If I recall correctly, however, the Athlons only come with the "quiet" HSF (aluminum block, no heat-pipes, quiet (slow) fan), like my Athlon X2 5000+ and Athlon X2 7750. Don't know if they changed it for the Athlon II series. If they haven't, then there really wouldn't be much noise - the drawback of course is that it may not be able to support the overclock (thermal limit).
And how come good air circulation in my case would change that?
Could you recommend some budget cases that meet those needs? Antec 300?
Good circulation means stale/hot air is replaced quickly and consistently with fresh/cooler air from outside the case. The result is lower ambient case temp, which also affects your CPU temp, and thus can also influence how high the fan needs to run.
On that note, Antec 300 is a great case. Probably the best case in its price range. I can also think of some CoolerMaster / Thermaltake budget cases (Elite line / V3/V4) which are all cheaper (and smaller) but can be competitive, especially if you don't require any bigger than they are. In the end though, budget permitting, I'd rather take an Antec 300. It doesn't hurt to invest a bit on the case, as this is one part of the system that can last a very very long time, surviving several upgrade cycles. In the long run it's better to get it right the first time.