Probably the easiest way to split tone is using Lightrrom. First I added a levels layer bringing the end off the histogram in to 20 & 232 to add a bit more contrast before importing into Lightroom.
I decided that I was going to tone the shadows a greeny/grey and the highlights a sort of mauve colour. In Lightroom head to the develop module & if your image is still full colour, convert it to grey scale with one of the presets or a mix of your choice in the HSL/Color/B&W panel. The then scroll down to the split toning panel immediately below, & this is where the fun starts.
Initially everything is set to zero
splittone panel 1.jpg
The top two sliders affect the highlights, the bottom two the shadows, & the centre one the balance between the two. Its a question of playing with them until you achieve the look you like. NOTE when you start adjusting the hue, you'll not see anything until you add some saturation. I started with highlight hue 290 saturation 40, shadows hue 105 saturation 40 but found that was a bit strong in the highlights for my liking & finally settled on these settings
splittone panel 2.jpg
The final image is attached