P.P.,
I think you are quite right - ordinary TV / FM tuner cards do not receive the frequencies (> 108 MHz) or have the steps (25 KHz or less) required to access aviation transmissions. Further, I need to set my receiver to AM rather than FM to pick up ATC here in the UK.
So Sholva, unless you want to build / modify your own PC card, you will need to buy a dedicated airband / multiband receiver. You don't need a transceiver - you really don't want to be able to transmit, especially if you are listening in to ATC at a busy airport!
Please bear in mind that in many parts of the world (certainly the UK) it is actually illegal to listen in to such transmissions (and this is also true of the emergency services), unless you are a legitimate intended recipient. Or it has been put in the public domain - e.g. the live internet feeds you mention.
There have been various discussions (mainly in Private Flying forum) over the years on this topic - who's to know, what's the harm, it helps with R/T, I've got an R/T license 'cos I'm a pilot so it's OK etc. etc., so perhaps you can search for these if you want to .
SD