Loading and displaying on VGA monitor a Background image in DE2-115's SDRAM
You need something like a video memory like the one found on casual graphic cards.

Change the VGA controller, so that it is reading the pixels repeatedly from the video memory. Update this video memory by the NIOS-II program according to your game, e.g., by copying the bitmap from the SDRAM to the video memory at the desired location.

The video memory will require two ports, one for the VGA controller and one for the NIOS-II. For the first step, you can use the one-chip block memory of the FPGA which offers two ports. But the available memory is rather small (just ~0.5 MB on the FPGA of your board), so if you start with 640x480 then you would have only few bits per color per pixel. Another option would be a video memory with a resolution of 320x240 pixels and then outputting each pixel four-times to get 640x480. 

For higher (color) resolutions, you typically need external SRAM (but only 2 MB available your FPGA board). But if this is already occupied by the NIOS-II, you can statically allocate some part of it for the video memory. To allow reading from the memory by the VGA controller, the bus connection the NIOS-II with the memory must be extended to some sort of multi-master bus.

Further notes:
To upload the image there are several possibilities: SD card, Gigabit Ethernet, RS232 port.