You can easily and simply buy an RCA ground loop isolator which (in conjunction with the RCA - 3.5mm cable) should solve the hum. Really we need to know what connections the TV has and similar for the stereo kit. You can purchase an analog - digital converter but that's overkill really and my not resolve the buzz/hum. I would also suggest trying a different RCA - 3.5mm cable as that's often an issue, preferably of a different design to ensure it seats differently (hopefully properly).
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[long answer alert]
I've been doing 'good' AV on a budget for a while now and would advise a slightly different route. As a side note, I know 'good' is a perception but when I refer to that I mean a reasonable amount of volume (enough to bother the neigbors - not that I intend to), sufficient quality to hear fairly accurate sound reproduction (I'm not talking audiophile levels - just roughly level tonal response without crackling) and (for me) decent seperation in terms of sounds in my 5.1 system being placed roughly where they should, especially when using DTS input like blu-ray or similar.
Sadly, as you've already purchased the Bose system, you've kinda started on that road and it's immensely difficult to get that to play ball well with an AV reciever. Bose stuff is designed to be a closed ecosystem and whilst the quality is high, the amps and speakers are highly tuned to each other. Removing their speakers and attaching them to the AV reciever would likely give a poor sound. I do say this without intending to upset or provoke and I do love Bose kit for clarity and reproduction - it is a 'good' system. I would relegate it to other duties, return it or sell it.
In your situation, I'd buy a used 5.1 AV amp from eBay to start. You have $300 and should be able to get something well within the bounds of workable for half that or just below, particularly if it's local and for pickup only. Ideally? Something 2-5yrs old with 3 or more HDMI inputs, 5 direct speaker outputs, a subwoofer pre-out and try to get something from someone like Yamaha, Sony, Onkyo or similar. Try to find one where the rear channels are not considerably under-amped as this will make a big difference with movies. Add some used 'eBay special' floorstanding speakers and shell out on a decent-ish used centre speaker with the rest of your budget. Throw in a cheap used active sub and you've potential for something 'good'.
My setup is static and similar to that. I upgrade stuff slowly as it starts to annoy me. The amp was changed from a 10yr old $150 to a fairly new $250 unit three years ago as the seperation wasn't perfect and the first amp didn't have HDMI. The rear speakers just after as my old ones didn't offer a straight enough frequency response (lacking in bass response) and most recently I've changed the sound card in the PC that's connected so my DTS/ProLogic signals are passed through properly rather than being 'downmixed' to two channels by the PC and upmixed by the amp 'guessing' where the sounds should go - this has given a massive improvement.
If you're looking for something 'good' but want to stick with 2.1, you'd be fine to do as above but leave off the rear speakers. Just don't try to mix the Bose and a non-Bose AV amp as this will provide very poor sound, plus with the Bose kit being quite pricey, would really be a massive waste of money! If you could get the AV amp to 'feed' the Bose, likelihood is that the delay in sound would be untenable - with lip-sync to the TV being a major issue.
Hope that doesn't offend!
