The better question is "What's more valuable to you: Your Time OR Your Money?"
While building your own PC is more cost effective, it's a major time sink. You have to buy all the parts, wait for them to arrive, assemble them & then the "fun" of installing the OS, the drivers & everything else to make sure it runs properly. If you're not tech savy & not great at personal troubleshooting, this will likely lead to a lot of headaches as you will not have anybody to assist you for technical support... or at least without paying ~$60 / hr for professional help. There's also all the wires & stuff to lug around, so it won't be very portable.
The Alienware Alpha is pretty much a ready-to-go platform that will have you playing your smaller Steam titles within 2 hours of opening the box. HOWEVER, I HIGHLY RECOMMEND getting a USB Keyboard + Trackpad / Mouse combo (like the Logitech K400r) to use the Alpha as a regular computer. Hardware set-up is very easy as it's only 3 cables (Power, HDMI & the USB 360 wireless dongle) & you just need an HDTV with an open HDMI port (or you can utilize the HDMI-passthrough, if needed) & takes less than 5 minutes to set up. The Alpha is very portable as I was able to fit mine into an old PS1 console bag with 2 controllers to take with me.
As for performance, this is where the standard "Your mileage will vary" as the Alpha tends to run on the "Good Enough" philosophy with a minimum average of 30 fps with default settings on some of the more graphically intensive games like GTA V. I have the i7 configuration myself & I've increased the graphical settings from defaults on GTA V to keep the frame rate at least 30 fps @ 1080p. While I won't be able to promise the same performance with the i5 model you're looking at, you can probably similar performance with default settings & possibly better if you opt for 720p (if your TV / Monitor doesn't support 1080p OR you actively want to run at the lower resolution). If ordering online from Dell (who owns the Alienware brand), you'll generally receive it in a week w/ standard shipping. Just make sure you take SALES TAX into consideration.
The case you've selected may be on the large side for the Alpha (as it's smaller than the XBox 360), but seems reasonable as well.
The one DOWNSIDE is that the CPU in the Alpha is non-upgradable... so if you may start running into potential performance issues around 2-5 years if you want to go with the latest & greatest titles. Everything else in the Alpha can be upgraded with standard laptop components.
While the choice is up to you, I went with the Alpha because I didn't have to time to invest in building my own PC & didn't have room for second full PC. Additionally I wanted something that was geared for gaming amongst my Steam library, which the Alpha fit my needs. I opted for the i7 configuration since I was able to secure a discount through my employer, otherwise I would have picked up the i5 configuration myself. I'm also a person that will run a system for 10+ years as well (assuming no critical component fails).
I believe you'll be happy with the Alienware Alpha i5 in the relative short term (like the next 2 years) & for the compact design that makes it travel friendly... but the if you're willing to put in the effort, the custom pc will provide longer-term happiness. Hence the original question that I've proposed above: Do you value your time (& go with the Alienware Alpha that you can get in a week or less) or your money (& build your own PC within a month or so)?
I hope this sheds some light on the subject so you make the decision that you're happy with.