I watch Eurosport via the ES Player app on a tablet (Nexus 7 2013) - picture quality is excellent in my opinion. Also, if you sign up for a year then it is £2.99 a month, well worth it for the season even though there's not much I want to watch over the winter.
As for hooking it up to a TV, that depends on what device you are using and what TV, so you'd need to investigate it yourself. As far as I know the app doesn't currently work with something like Chromecast, don't know if there are any plans for this. The feed is in HD so I would expect it to scale up fine, but I haven't actually tried so wouldn't want to promise anything.