The fastest way to accomplish this is to use an OpenType font that includes the glyphs you need, but failing that, the next-fastest way, if you're working with CS4 or later, is with a GREP style for each of the characters you are trying to replicate.
Set up the kerning pairs and Character Styles the way Lauren describes, with this exception: a Character Style doesn't accept kerning values other than blank, Metrics, Optical and None, so you have to do this with tracking rather than kerning. (You'll probably need different tracking values for different letters.) Then, in your Paragraph Style(s), set up a new GREP style for each combination of letter+ogonek to apply the appropriate Character Style.
From then on, InDesign will automatically apply the GREP style every time you have a letter+ogonek combination in the text. You can watch it happen on the fly, as you type!