Looking for Help Building a Streaming/Gaming computer

That's a REALLY big budget range there. You also might want to be a bit more specific about which games you'll be playing.

Alright first question of the day do you want your system to be compact or a beast? My system right now is as big as you probably want for a PC just because I have it water cooled.
Well for recording gameplay I would immediately go with Nvidea GPU because there shadow play is amazing I use it all the time. If you don't know what that is its nvideas game recording tech that is built into all kepler GPU's and it uses almost no gpu power to record so you don't lose fps on games.

Since I don't know if you want a compact or beast pc yet I will give recommended inerts
either i7 4770k or the i7 4930k that is a 200$ differance for 4 more cpu threads wich will help with video editing if you need that otherwise if you are just going to be streaming video id stick with an i5 4670k because for just gaming you do not need that much cpu power but i5 is the lowest you should go.

Motherboard I would go with this one if you go with the i7 4770k or the i5 4670k http://www.newegg.com/Product/Product.aspx?Item=N82E16813130692 it is a great gaming motherboard and it has most of the features of an amazing asus mobo or if you get the 4930k id go with this mobo http://www.newegg.com/Product/Product.aspx?Item=N82E16813188130.

Ram I would go with this http://www.newegg.com/Product/Product.aspx?Item=N82E16820231461 I love gskill and this is low voltage wich will save some power and its still 1600mhz which is plenty for gaming oh and 8gb is also plenty for gaming and streaming.

Hard drives I would deff go with either a 120gb or a 240gb ssd for a boot drive and than raid 0 two 1 or 2 tb hdds. I would go with this as the ssd for boot http://www.newegg.com/Product/Product.aspx?Item=9SIA2W01AZ5378 or http://www.newegg.com/Product/Product.aspx?Item=9SIA2W01AZ5378 and for your hdd's id go with 2 of these http://www.newegg.com/Product/Product.aspx?Item=N82E16822236624

GPU I would go with as long as your streaming/playing in 1080p and nothing higher a single 780ti would be great if going to 1600p or 4k I would definitely go for 2 780ti's

PSU I would definitely go for the highest wattage you can get because if you are going to get 2 780ti's and maybe in the future get another or something than you are prepared I love Seasonic they are amazing but corsair is also great because Seasonic makes the inerts of there psu's. id go with either this one http://www.newegg.com/Product/Product.aspx?Item=N82E16817151109 or this one http://www.newegg.com/Product/Product.aspx?Item=N82E16817139014

I did not give the CPU links or the GPU links because prices change with those quickly these days so I would research them yourself and also if you need me to go into detail about anything I said just say and i will alright.