I assembled my PC back in '07, with a quad-core Q6600 (like an i5 from four generations back), a couple gigs of (DDR2) RAM, and a fanless nVidia 8600 GTS (I prefer my computers quiet). Played all the Dragon Age and Mass Effect on it, and a few other things; not at the monitor-native 1920x1200, but something lower like 1280x800 or 1680x1050 with the graphics card scaling it up. A little fuzzy, but not that noticeable if you're not doing A/B comparisons. Over the years, put in an SSD with 64-bit Windows 8(.1) and brought the RAM up to 6GB.
Last year, bought Tomb Raider when it came out, knowing that I would want to upgrade the graphics. Never got around to it until the beginning of this month. (If I had waited until today, Steam is having a 75% off sale -- would have saved another ten bucks.) Turns out nVidia recently released their new Maxwell products, headlining with the mid-range GTX 750 Ti. The thing with Maxwell is performance per watt: the cards are small (normal sized, not extra-long) and don't require auxiliary power. With the "SuperClocked" (slightly faster) one from EVGA (with 2GB of video-RAM) I average 67 fps at 1920x1200 on High quality in the Tomb Raider benchmark (V-Sync off).
Midway TL;DR: if you have a desktop and don't have a high-end card from the past two or three years (in which case you're probably set already), a GTX 750 Ti is a no-brainer for about $150. Good performance that will fit (physically: size and lack of extra wiring) in most systems. You may not be able to play DAI at 1080p with all the dials turned up, but it should still look pretty good at a decent frame rate....
As long as you meet the other requirements of course. For Battlefield 4, which uses the same engine, required RAM is 4GB, 8GB recommended. Not sure how that will affect game play (maybe load times), but most people have at least the minimum. Same with OS: most people have 64-bit Windows 7 or 8; latter is recommended. (The required OS for BF4 is 32-bit Vista, which means you can't even use all 4GB of RAM; you need a 64-bit OS.) A little more complicated with CPU, but dual-core minimum, quad (Intel, hex AMD) recommended: again very common. For a lot of the people with desktops posting here, the graphics card is the main variable. For laptops, you will need a recent high-end discrete mobile GPU.
Data point: last year's Razer Blade laptop with the nVidia GTX 765M gets 44.7fps at 1080p on Tomb Raider High, about two-thirds what I got with the 750 Ti. On a sort-of-plus side, if your laptop's native resolution is lower than 1080p (or an integer multiple of it), you will want to game at a lower resolution to get better frame rates anyway. (They got 57.5fps at 900p.)
Final TL;DR: you may not have to upgrade your whole desktop rig to play the game. A simple graphics card update may be enough.