Many players will find and play the subtitle file if it is in the same folder as the avi video file. It needs the same title as the avi file, but with the appropriate file extension, in other words, avimovie.srt or avimovie.sub, depending on what type of subtitles you have.
You can burn in subtitles permanently using lots of different tools. AviDemux can do this if you reencode the video and use one of the subtitle filters.
Does it have to be in avi format? Other container formats are better with subtitles. MKV or MP4 would work with a selectable subtitle stream. You could use Mkvtoolnix (mkvmergeGUI within that set of tools) and input your avi file and a subtitle file and produce an mkv file with both.