There are multiple solutions (Roku, Boxee, etc.) but Chromecast and PLEX would do it the cheapest. Wireless you're pretty much limited to 720p no matter what solution you use, so Chromecast/PLEX is probably your cheapest and easiest choice, with the added bonus that you can keep the files in any quality you want and let PLEX convert them to 720p.
If you absolutely need 1080p, you'll have to run wires or bring the files physically to the TV somehow. Regardless of resolution demands, depending upon which TV you have, there is a chance it could play the movies directly from the HD via USB. If that doesn't work (which it probably won't, as typically you'd need to have formatted it FAT32, among other complications), you could copy the movies to a USB stick if it will play the formats (or convert what you have to something it will play). I am assuming you don't want to do any constant moving of the HD or copying to USB sticks though, which leaves you with wireless.
Something like a Boxee or Roku would cost more, but I know you don't typically worry about a few hundred bucks, so it would depend on what you find most comfortable to view with. Chromecast is going to require you to use your phone or tablet, whereas the others will have remotes of varying effectiveness.
Sidenote: I'm talking an older Boxee Box, not the new one (I forget the exact name). The new one is practically worthless to be honest, so you'd have to find an old Box as leftover stock or used or something. Never used a Roku, but that's probably easier to get now and supposedly just as good or better.
There might be better solutions depending upon your TV (I believe Samsung has some sort of native streaming thing in their new TVs) or something I'm forgetting, but Chromecast for $35 + $5 PLEX app on your phone is going to be tough to beat I think.