Is it possible to use a raspberry pi to control these LED strips?
For DMX512 you need a UART capable of 250000 baud and 8 data bits with two stop bits (or 8 data bits a space for parity and one stop bit) 

I tried a cheap* RS485 dongle and while it claimed to manage the necessary framing minicom couldn't make it do 250000 baud.

I'm not sure what the exact capabilities if the Raspberry Pi main UART are.

(*)it cost less than $1

it looks like FTDI uarts can do 250000 Baud so one of them and an RS485 interface chip should probably get you there. 

You can of course buy DMX adaptors that connect to USB or to GPIO but shopping questions are off topic here.