I’d recommend a AMD 6-core AM3+ CPU for $120, a $60 motherboard to support it, a $140 pair of 8GB DIMMS (so 16 GB total) and a $150 GTX 750 Ti graphics card. Add a $75 128G HyperX SSD and a $50 power supply and a $50 box and you’re good to go for a total of $650 or so. If you have slightly more budget, look for a 760 and then a 770 graphics card, then a bigger SSD, then Intel instead of AMD, in that order.
This is assuming you either run Linux (bleeding edge) or, more likely, have a copy of Windows that you can re-use. Else you’ll have to add one of those, which adds to the cost. (You may be able to look into the BizSpark program to get cheaper Windows and Visual Studio tools, if that’s still going.)
Some people will say a 120 GB SSD is “not enough,” but if your main thing is surfing the web and running UE4, then it’s fine, and SSD is much, much better than spinny disks, so well worth the sacrifice. If you do lots of torrents, video editing, installing big games, and file hoarding, then 120 GB is obviously not enough; perhaps add a secondary spinny drive for bulk media?