Help getting started with Roku devices, including setup, connecting your device to your TV, linking a Roku device to your account at my.roku.com/link, adding channels, subscribing to services, and more.
It isn't clear if your older TV has a DVI or DisplayPort connection. So, you'll need either an HDMI to DisplayPort or an HDMI to DVI adapter.
If those assumptions are correct, you're now looking at weighing the cost of things.
Roku Streaming Stick+ is $50 (though you can often find them on sale).
HDMI (female) to DisplayPort adapters run from $15-40.
HDMI (female) to DVI adapters are around $30+ and you still need DVI and audio cables.
Where all this is leading is to consider the cost to do what you're seeking vs the cost of a similar size Roku TV. Of course, cost isn't always the most important factor, but it may be worth considering.
DBDukes http://www.dbdukes.com/Roku Ultra (4660) Roku Streambar (9102) Roku Streaming Stick+ (3810)