With a budget of $850 or less you are not going to get a quad core CPU unless you want a laptop with an AMD APU. However, each AMD APU core have less processing power than Intel CPU cores so when only 1 or 2 cores are being used by a program the Intel CPU will have no problems outperforming a quad core AMD APU. AMD APUs also uses more power compared to Intel CPUs so they will provide less battery life.
Below is a link to some information regarding the AMD quad core A10-7300 based on the current generatin Kaveri APU. The most relevant benchmarks to you are the X264 HD Benchmark 4.0 (Pass 1 & 2); click "Show Comparison Chart" to see how it performs against other APUs / CPUs.
http://www.notebookcheck.net/AMD-Kaveri-A10-7300-Notebook-Processor.117333.0.html
For video encoding purposes the A10-7300 APU is more or less equivalent to a 4th generation Core i3 CPU. That means it will have lower performance compared to a laptop with a 5th generation Core i5 CPU especially when you take into consideration the Core i5 CPUs have Turbo Boost that increases the CPU clockspeed as long as there is demand for the CPU and it is not too hot.
For the moment I recommend two slightly different laptops to you; the 14" ThinkPad E450 (Intel) and the 14" ThinkPad E455. Both start at 4lbs and their base configurations are relatively inexpensive at $550 and $427; respectively. That leaves you room for some upgrades. Superficially, both laptops are identical, the only differences are the motherboards and processors. Lenovo claims battery life of 6.5 hours for the E455 and 8 hours for the E450.
ThinkPad E455 - $427 base
http://shop.lenovo.com/us/en/laptops/thinkpad/e-series/e455/
- Upgrade to the A10-7300 which is the most powerful APU for the laptop. Doing so will automatically install a dedicate graphics chip (Radeon R7 M260DX). This ups the price to $607.
- If you do not play games, then opt for the quad core A8-7100 APU which is only 100MHz slower than the A10-7300. A slower dedicated graphics chip will be installed (Radeon R5 M240). Total price is $560.
- I recommend not upgrading the RAM from 4GB to 8GB since it will cost you $80. You can purchase a 4GB stick of DDR3L 1600MHz RAM for $40 or less and install it yourself.
- Upgrade the 500GB hard drive if you wish.
ThinkPad E450 - $550 base
http://shop.lenovo.com/us/en/laptops/thinkpad/e-series/e450/
- Upgrade to the Core i5-5200u which can Turbo Boost up to 2.7GHz on a single core and probably 2.5GHz when both cores are used. This will bring the price up to $645 which still gives you some wiggle room in your budget.
- Do not upgrade to the Core i7-5500u since it will add a dedicated graphics chip (Radeon R7 M260DX) and bumps up the price to $902. Unless you really want it.
- This laptop model gives you an option to upgrade the screen from 1366x768 to 1920x1080 if you wish. Personally, I find laptops with 1366x768 resolution screens to be useless. But that is just a personal preference. This will bring the price up to $721.
- Again, do not opt for additional RAM since you can do it yourself for $40 or less.
- Upgrade the HDD if you wish.