For this specific illusion spell fitting your criteria (I think) it is clear that the caster still experiences the illusion (PHB, p. 264):
Casting Time: 1 action
Range: 60 feet
Components: V, S, M (a bit of fleece)
Duration: Concentration, up to 1 minute
You craft an illusion that takes root in the mind of a creature that you can see within range. …
While a target is affected by the spell, the target treats the phantasm as if it were real. The target rationalizes any illogical outcomes from interacting with the phantasm. …
The victim of a Phantasmal Force will experience the illusion even when using the familiar's senses. It "takes root in the mind of a creature" so it does not matter what senses it uses. Even if illusions are not seen when using the familiar's senses in other cases (yet to be discussed or established in another answer), for this spell "the target rationalizes any illogical outcomes" so they will believe it and experience it anyway ("My familiar can't see the invisible orc!").
(Edited) Other illusion spells
I've quickly looked though all the illusion spells in the PHB and they pretty much all create an illusion that you sense from whatever point of view you are looking from, so you would sense them from your familiar too.
However if your familiar has a sense that the illusion does not cover then your DM may give you a save, perhaps even at advantage, if the environmental factors warrant. For instance a bat's blindsight definitely won't sense a Minor Illusion if it is a visual illusion. However you don't get a save unless the DM says so: maybe your character has other possible explanations that cover the situation (e.g. it's possibly because of a silence spell effect) and it doesn't tip into proof or interaction with the illusion in a way that possibly reveals it for what it is.
However there is one clear exception, Phantasmal Killer, which only effects the target. However this works at least in part like Phantasmal Force so "looking" through the senses of your familiar may not help. Again the DM will judge if the save you get every round is effected by this extra information. I've asked a question about Phantasmal Killer to see how closely it should work to Phantasmal Force: Does Phantasmal Killer work on a blind target which discusses things pertinent to attempting to view the spell effect through something elses senses.