Poor Man's 2 channel HT--advice sought

"Can I, for example, get a pair of PSB Alphas or Paradigm Atoms and drive them from my TV?"

Does you TV has speaker connections/hookups in back, i.e. is there anywhere to hook-up the speakers. Some TV's actually do/did have hookup for another pair of speakers. Most don't so you probably can't run speakers of your TV short of taking the thing apart and disconnecting the wiring from the internal speakers and then, after crimping/soldering on new wiring run the sound out to the new speakers. How much power does your tv produce, 1 watt, 5 watts? Probably won't work well even though it can be done.

"Or use the TV's audio outputs to drive an old Harman Kardon receiver I have and use that to drive the speakers?"

This might work. Are those audio outputs RCA line-level. The real hitch will be if they are a fixed level signal or if the volume from your tv will adjust the signal. If its the former you'll have to adjust the volume from your H/k receiver (might be a bummer if you want absolute convenience), the latter and you can adjust the volume from your tv and have the volume knob on the HK fixed/set level.

"I want to spend next to nothing on this project, by the way."
For the speakers either buy used, buy off UBID or Ebay, or DIY. You will have to decide just how much fidelity you want. You could build a nice pair of DIY speaker equivalent, if not most likely better, to the ones you mentioned for <$100. Or you can always find something for as little as $20pr on the auction block.

Cheapest fix I can think of; pick up a hifi vcr--for the tuner--then a used, older receiver. both of these could be had near 100 ea. then some speakers. The tv signal from a cable box or tv antenna goes into the vcr--then into your receiver--into the speakers. The audio from the dvd player--pcm mix-down--also goes into the receiver. I'm sure there are plenty of good older receivers to be had for100.

If you have an old integrated amp (Harmon Kardon) Run the audio signal directly from your DVD player to the pre-amp and out to the speakers. Take the video signal from the DVD player directly to the TV assuming you have a video input on the TV? I wouldn't go so far as to say this is a dumb idea, just the first step on your way to H/T Nirvana.