How i could find how program existing board with atmega chip on it
How could I find out how to program an existing board with an ATmega chip on it?


It is likely that the toy maker has provided a proprietary system for programming the toy. Reverse engineering that is out of scope here (at least it isn't something I'd spend much time attempting). However you can probably adopt a more generic approach as follows:


Look up the data sheet for the CPU chip


ATMEGA16L 

Learn about the programming interface for that chip
Locate the appropriate programming pins on the IC and how to connect
to them - look for programming pads (used with pogo pins) or headers
(usually 3x2 or 5x2 0.1" header block for Atmel)
Work out how that chip has been wired up to the devices you want to use it with
Ensure there is no usage clash with pins needed for programming - or work out how to isolate those pins from other devices
Download a suitable program development environment


WinAVR and AVRdude
Arduino IDE (if your chip is supported and circuit is compatible)
Atmel studio
etc

Obtain a suitable programmer


AVRISP
USBASP
Pocket AVR Programmer
Arduino as In-System-Programmer (ISP)
etc

Connect the programmer, easy if there is a programming header, 
harder if you have to solder wires to pads or carefully to IC (especially hard for smaller SMD IC packages)
Learn the appropriate programming language
Use the IDE to develop a small test program and upload it to the CPU.


Note that doing this will likely overwrite any bootloader the maker programmed into the CPU and may therefore render it unusable with the original toolset provided by the toy maker.



References


LadyAda - AVR Tutorial
Atmel - Getting Started
LadyAda - Choosing a programmer (a bit old)
Hackaday - AVR Programming intro