the big question:
why is your onboard poor quality?
if you are getting buzzing, crackling, distortion, poor audio quality or other issues you can hear from your onboard an amplifier is NOT going to fix that. to fix those sort of issues you would need to buy a soundcard or use an external DAC+AMP.
just so you know, an amplifier is only meant to boost the power of a signal so that you can power harder to drive headphones with a device that has a weak output. an amplifier would be the correct thing to think about if the problem you have is that you have weak output but good quality (low volume levels but good sound quality).
the fiio e07k is a dac-amp so would work. the e10k is a dac-amp as well. the e11k is just an amp so would not fix any sound quality issues.
be aware: you will only get 2.0 sound out of the headphones. if you plan to game with them (and use virtual surround) you might want to go with a soundcard instead of a dac-amp. the xonar dg is a good budget choice and at $27 beats them in price as well.
i am not too familiar with super budget level headphones under $70 although i will say that the hd518 is an excellent choice for $75 (uses the same drivers as the more expensive 558 and 598) and is worthwhile going with for the price. if you're into gaming and wanted something with a wide soundstage the ad500x is a good choice although its going to have almost no bass (it has mid bass but now low bass) which makes it easy to hear footsteps (advantage in fps) but a bit less fun sounding overall.
as far as lower cost... i'm aware that the creative aurvana live headphones get good marks around $50 although they sound best at low volume levels. i know the hd429 is considered good for heavy bass and bassheads. head-fi has a few posts and theads about low budget stuff that i'm more unfamiliar with in case you absolutely needed something cheap.
my own recommendation would be the hd518 + xonar dg.
the 518's are fairly neutral with a slight bit more bass than true neutral, nice wide soundstage and a laid back mellow sound to them which is nice for easy listening and long sessions. comfort is about average but most find them perfectly fine. the xonar dg is cheap at $27 and has an amplifier just as - if not more powerful than the portable solutions you're looking at (not needed for the hd518!) as well as a DAC which is better than low end onboard and equal to or slightly better than high end onboard. as long as you have a spare slot inside your pc, it would work fine.