Preheat your oven to 425 degrees.
Slightly grease a baking pan, or use a piece of parchment paper.
Place flour, butter, and cream cheese (torn into small pieces) into a large mixing bowl.
Quickly use your fingers to partially incorporate the butter and cream cheese into the flour. (You want to see some small pieces left after the mixing process is done. Don't overmix; you want a consistency between cornmeal and shaggy.) Mix in the chives and cheddar cheese.
Make a well in the center of above mix, and pour in the buttermilk. Now, it is crucial not to overmix your dough. You don't want visible flour pockets, so as soon as all is incorporated, stop.
Transfer the dough from the bowl to a well-floured surface, and lightly dust the sides and top of the dough.
Pat into a smooth shape, and use a rolling pin to get the desired thickness (about 1/2 inch). Have a small bowl of flour nearby.
Dip your biscuit cutter into the flour, and start "stamping" your biscuits, and placing them side by side — touching — into your pan. Be sure that your biscuits touch, so they will rise up, not spread out. The key to good biscuits is to not have to roll your dough out too many times.
Brush liberally with melted butter, and place in preheated oven for 15 to twenty minutes, rotating the pan once, halfway through. Every oven is different, so the first time, you need to watch them more carefully. You may need to adjust the temperature 10 to 25 degrees, or the time to 25 minutes.