How do I get my late-2012 MacBook Pro to work with my Seiki 39 inch 4k monitor?
You will need an active adapter that converts the DisplayPort 1.1 signal coming from your Thunderbolt port to an HDMI 1.4 Signal. The UltraAV Mini DisplayPort 1.1 to HDMI 1.4 Active Adapter for example. The regular (passive) adapters will not give you the HDMI 1.4 that is required for your resolution.

Also, make sure your HDMI cable is High Speed certified (sometimes it's called a Category 2 cable), and that you use the right resolution, refresh rate and color depth.

HDMI resolutions

HDMI Standard    1.3a                 1.4                   2.0 
HDMI cable       Standard/Cat1        HighSpeed/Cat2        HighSpeed/Cat2
Resolution       2880x576  2560x1600  3840x2160  4096x2160  4096x2160
Max Refresh Rate 50 Hz     75 Hz      30 Hz      24 Hz      60 Hz
Max Color Depth  48 bpp    30 bpp     36 bpp     48 bpp     30 bpp
                                      ^


Mind the maximum color depth, although 24 bits per pixel should suffice for most uses. On the Mac, 24 bit is sometimes called "millions of colors".