You will need to create what is called a mezzanine board. Your big problem is that the 5565 only needs ~ 25 signal pins (13 address, 8 data, 2 enables, 1 R/W, and 1 output enable), while an 8k x 8 dual port RAM needs at least twice as many. This means that your DPRAM will have a much larger (in terms of pins) package than the 5565.
A good starting point for DPRAMs is the IDT 7005, which can be found on eBay. This will come in various packages with 64 - 68 pins.
First, your game board will need to be modified to allow connection to the 5565. This can be done either by unsoldering the RAM and replacing it with a 28-pin DIP socket, or by soldering connector points to the RAM to connect with the new board. Replacing the IC with a socket is the better way to go, and I'll assume you take that route.
The mezzanine board will consist of 5 parts,
1) 28 pins sticking down to mate with the 5565 socket
2) a DPRAM with one side (call it side A) connected to the 5565 signals (with slight exceptions - see 4 and 5)
3) an interface connector to allow side B to be connected to your MCU, and
4) a small amount of logic to convert the 5565 chip enable lines to be compatible to the 7005. The 5565 uses 2 enable lines, one active high and the other active low, while the 7005 uses a single active low enable line.
5) some logic to indicate to the MCU that the game is writing to the same address that the MCU is trying to read. This can get tricky. I'd advise setting up address registers and an output register and not trying to read the 7005 directly, but rather slowing your MCU read cycles down enough to allow contention to resolve.
Oh yes, and be careful about how long a cable you use between the MCU and the board. You don't want ringing on your address lines. Series termination will be a good idea.
Best Answer
You can get 512MB for $10 and they even throw in a complete ARM computer around it.
If you really want a serial RAM to retrofit on an existing board, the irony of modern ultra-mass-production is that it would probably be cheaper to stick the above ARM SBC with a little bit of code on it to emulate a serial RAM than to purchase the dedicated hardware device (if it even exists). And while you're at it, replace the existing micro with it...
If this is for DIY or low production volumes, I suggest you use such a module. Otherwise, there are Cortex-M4 or M3 micros with SDRAM interfaces, so you can simply add a cheap SDRAM chip.
The drawback of SDRAM and a big cpu is the higher power consumption versus SRAM and a slow cpu. If you absolutely need very low power, then that will be an issue. But SRAM is a lot more expensive and less available than SDRAM.
Here's another non-volatile option but also too small and expensive, although these non-volatile chips offer an interesting option: power draw is very low when not in use, because you can power them down!
What kind of speed and power do you need?
If this is for data logging, and the data throughput is low enough, the number of writes low enough, you could also use a SPI flash...