How do I calculate the DC for a spell cast by my familiar via the familiar spell feat? (i.e. Do I use my ability score modifier, or the familiar's?)
It is unclear
The feat, as written, has a few inconsistencies. Here are the things to note on the feat:
Variables that rely on caster level function according to your caster level, not your familiar's Hit Dice, though your familiar's Intelligence may influence how precisely it can follow your instructions on how to use these spells.
Your familiar must be carrying any material or focus components necessary, unless the materials cost less than 1 gp and the spell is prepared with Eschew Materials. A familiar spell counts against the number of spells you may prepare for as long as your familiar retains the spell.
Well, first, the mention of caster level would be entirely unnecessary if the feat simply said that you are casting the spell, but the familiar used her action to activate and deliver it, similar to the Deliver Touch Spells familiar ability. Or, instead, simply said that you store the spell in your familiar to be used at a later time. Because the caster level was explicitly called out, now we have the confusion about other variables ("what about the DC?"). Not to mention bonuses to caster level from feats (Spell Penetration)...
Secondly, why would the familiar need to be carrying material components if they are not actually casting the spell? They could have saved space (and confusion) by simply said that you must provide any materials when the spell is given/stored in your familiar. Because they must be carrying expensive materials, this also implies that they are the actual caster of the spell.
As written, everything in the spell indicates that you should use your familiar ability scores to define every undefined variable, such as DC for saving throws (which includes things like Spell Focus), or touch attack bonuses (which includes the bonus for being tiny or small).
Finally, I could not find any developer clarification so that we could tell you how the rules as intended are supposed to work here.
So, my advice, considering that this feat is not strong by any means, would be to allow the familiar to use your ability scores, feats and similar abilities that do not rely on using an action to be activated (like Conceal Spell), to calculate the spell effect. This would eliminate the confusion, as you won't have to calculate anything on the fly, and the character is not penalized for taking a feat.
You cast the spell, so you will use your spell DC for the spell (at the level of the spell before applying Familiar Spell).
Variables that rely on caster level function according to your caster level, not your familiar’s Hit Dice...
While CL doesn't determine DC, you can tell that it's still the PC that's casting the spell, not the Familiar. Furthermore, the Feat specifically calls out the differences from normal casting.
Your familiar must be able to speak to cast spells with a verbal component (the ability to speak with its master or creatures of its kind is insufficient). Your familiar must be carrying any material or focus components necessary, unless the materials cost less than 1 gp and the spell is prepared with Eschew Materials. Attack rolls use your familiar’s ability scores.
Because it doesn't list a change in the DC based on Familiar statistics, the DC does not change from otherwise casting the spell.