There are a number of options out there such as the AverMedia Live Gamer Portable Lite shown above. You can get this for around £60 and boasts video capture at full 1080p. That said most options on the market also capture at 1080p but you can pay upwards of £100 to start for some. Having had personal experience with the AverMedia, I can't fault what you get for the price with HDMI pass through and audio options the quality is really good.
Making gaming videos is not as daunting as it sounds providing you have the right kit. There are options out there for capturing game footage without connecting to a PC such as the Hauppage HD PVR Rocket.
Another thing worth thinking about is what games are you looking to record? Retro games need far lower bitrate (how much data is saved per second) than games such as Call of Duty Infinite Warfare and it depends what you want to achieve, if it's the visual wow factor that your looking for then you will need the higher bitrate and the larger file size. But, be warned, video editing high res game capture is going to need a good a high spec computer ideally (we used a Mac Book Pro with the AverMedia with a quad-core i5 processor and 8 gig of RAM, the 128 Gig SSD got full pretty quickly though). Video editing software is also important, you can get free options, we used iMovie as that was already pre-installed but keep in mind that you may need to buy some decent editing software unless you look at buying a capture device with some bundled software.
Hopefully this has given a little background as to what you may need if you are thinking about doing some game capture. There are other things to think about such as setting up a YouTube channel for example and your audio requirements, but we'll look to cover these at another time.