The original file I received from the client color mode is CMYK, and I need to create RGB files from it, How can I do that in the most proper way, without facing any color problems or having to create the artwork from scratch ?
File > Document Color Mode > RGB
Then, if you need to adjust colors, you'll have to adjust them.
There's no simple easy method of switching color modes and maintaining exact colors in every instance. However, converting from CMYK to RGB should present little, if any, problems. It's when you covert from RGB to CMYK most color shifts occur. The RGB gamut is much larger than the CMYK gamut and the RGB gamut encompasses all colors possible in CMYK.