The A4 is a dual core cpu so upgrading to a used Quad core would make it close to the FX4300 or the other quad core apus at the same clock speed.
Finding a used FM2\FM2+ APU will be a challenge, and many will be selling them with a board and memory. CHECK YOUR APU SUPPORT LIST FIRST.!!!
OR...
Because you have DDR3 memory, if you find a used CPU+MB+Cooler that uses DDR3 memory that could be a better option. However doing a board swap could mean doing a OS system re-install. Maybe going with a PhenomII with a AMD chipset will allow for just some driver updates and re-installs. It's a gamble if it boots in safe mode and lets you fix it or not. So if your going to go with this option, make sure what you pick is faster than a Athlon II x4 or FX4300, otherwise it's not worth the effort.
I would look for used parts locally. Stick with CPU+MB combos because many of those old boards are hard to find.
1st place. Depending on what is for sale by far the Sandybridge\Ivy bridge i5's would be the fastest option that you will find used for sale.
2rd place is the socket 1366 i7 and Xeon.
3nd place is the Phenom II X4 or X6. Make sure it's a model with L3 cache and at least 3 GHZ. Also the FX8300, ect.
4th place is the socket 1056 i5, i7,Xeon and 1366 i5. Make sure it's not a dual core i5, there is a oem i5-530, ect. that are just a socket 1056 i3 with HT. FX-6300. Sandybridge\Ivybridge i3's with HT.
5th place is swapping out your A4 for a quad core APU or CPU (There were some athlon X4's for that socket)