as the @Shinrai suggest it is necessary to disable onboard graphics. i have also hd 5770 and intel dg41ty motherboard. my card was not working like you at first installation.
look for the option where you can choose between IGP(integrated graphics processor) and pci-e. it must be in bios i can guarantee.
steps:1. first connect the motherboard to onboard graphics port.
2. go in to bios and look for option to choose between the two option, chances are that you will find this option in chipset configuration tab or something similar on your motherboard.
choose the pci-e from that option. then save the option and exit bios.
3. do not restart just after saving the option. first shut down your pc by pressing the power button on cabinet.
4. now after shutdown connect the monitor cable to graphics card output port.
5. start the computer and now you should be able to get the display.
the same procedure i applied just 1 month ago.
you need to just look carefully the bios option of your motherboard. it is there for sure my friend.