I do not believe there is any way to get a homunculus companion. If there is, it's free to take the usual archetypes.
Homunculi familiars, like other Improved Familiars, do not get the "speak with animals of your own kind" ability, which is traded out by almost all familiar archetypes, so it can't take any of those. The exceptions are, uh, Emissary (which I wouldn't consider appropriate for a homunculus anyway), Sage, and School Familiar.
Oh, so it does. But it's not an animal companion, nor stated to be equivalent to such (in fact it's pretty clearly not), so I don't think the animal companion archetypes would apply.
What kind of homunculus are you looking for?
You might be able to convince your GM to let you use the rules on Building and Modifying Constructs to gradually build in extra abilities for gold. But the Promethean Alchemist homunculus is clearly different from normal ones, and doesn't even say it has construct traits AFAICT(!), so this may be a hard sell. Not every GM would allow it even for a regular familiar homunculus.