Will this video card run skyrim?

It's a pretty bad graphics card. It'd play Skyrim or Minecraft on lowest settings.

This is a inexpensive graphics card but it'll play Skyrim on medium settings without any problems - http://www.newegg.com/...6814127738

As the previous answer stated the HD5450 is a low end card that's not really designed for gaming and it won't run skyrim very well… However you may be able to run it at low detail and at lower resolutions ( eg 1024x768 ) … But at those settings the amazing world in skyrim will look pretty bad
minecraft on the other hand is a less demanding game and I know for a fact that the HD5450 will run this game ( as my mates got that card and runs minecraft pretty well ) … However garrys mod is a source based game and will run at medium detail
that said you would be better off spending a few more dollars and buying something like the "XFX HD-657X-ZDF2 Radeon HD 6570 1GB 64-Bit DDR3 PCI Express 2.1 x16 HDCP Ready Video Card "( http://www.newegg.com/...6814150613 ) . This will run skyrim at medium detail up to 1440x900 resolution and will run minecraft at medium and garrys mod at medium to high ( all at 1440x900 resolution )
at the end of the day its always best spending that little extra as you will get better performance for a small additional expense… And if your budget could stretch even further then the "SAPPHIRE 11192-07-CPO Radeon HD 6670 1GB 128-bit DDR3 PCI Express 2.0 Video Card "( http://www.newegg.com/...6814202074 ) would be even better… This would run the games at the same detail level but you could run it at 1680x1050 resolution ( which would make the game look better )
i've been building pc, s for over a decade and in my experience buying a cheaper card is wasted money… Once you run games with that card you will start to discover it struggles with even the most basic games, and within a few months you will need to upgrade again… So its always best to buy a slightly better card in the first place
I hope this helps calvin… Any questions let me know