Honestly this is touchy! First off some have modified their X51's to fit a GTX 970. But you would have to remove the front metal intake fan grill, Or find a different cooling solution for the CPU. And you need to know the wattage of your current power supply. Because your current PSU may not be powerful enough. The 970 needs a 500watt PSU of decent quality. I would honestly go with a new Case before I would go through modding a case that small. You can simply just move your X51's guts to a mid tower case. I wish I could give you better advice. Here is a video that might help https://www.youtube.com/watch?v=nVi9puLG00A

But if you attempt this be very careful! Remember static electricity from your hands can hurt electronics, So try not to touch any circuit boards. If you never done this before find someone who knows what they are doing. Or just go with a diffeent GPU option. If it was me I would just buy a new CPU case for my system and get rid of the small X51 case. I do not recommend using a GTX 970 in a case that small. A GTX 970 runs up to 75c and the Ram runs really warm too. So it needs good airflow. If you do get it to fit, you will more than likely have heat issues because of the low airflow.

The GPU goes inside your computer not your monitor. But if your asking if it will work your monitor? Yes it has both HDMI and DVI plugs. It even has a DVI-VGA adapter. However I have no idea if it will fit inside your computer. You need to list the specifications of your PC.

If I remember correctly the case used for the XPS 8300 should accommodate it. However be sure to replace the power supply that came with your Dell. The wattage of the factory PSU will be too low. And honestly Dell uses cheap power supplies. And the GTX 970 needs at'least a decent quality 500watt unit. Best Buy has a decent deal on the EVGA 500WB for $48. And they have a Thermaltake Smart Series 650 Watt (SP-650P) for around $60. Just make sure not to buy the Thermaltake TR2 Series. The Smart series is of much better quality. But the cheaper EVGA 500B unit will work fine. It's a good quality unit too.