The Trickery Domain cleric gains the Channel Divinity feature Invoke Duplicity:

As an action, you create a perfect illusion of yourself that lasts for 1 minute, or until you lose your concentration (as if you were concentrating on a spell). The illusion appears in an unoccupied space that you can see within 30 feet of you. As a bonus action on your turn, you can move the illusion up to 30 feet to a space you can see, but it must remain within 120 feet of you.

For the duration, you can cast spells as though you were in the illusion’s space, but you must use your own senses. Additionally, when both you and your illusion are within 5 feet of a creature that can see the illusion, you have advantage on attack rolls against that creature, given how distracting the illusion is to the target.

The forcecage spell description states:

A prison in the shape of a box can be up to 10 feet on a side, creating a solid barrier that prevents any matter from passing through it and blocking any spells cast into or out from the area.

Can I cast a spell that doesn't require concentration through the clone granted by Invoke Duplicity while I'm under the effect of forcecage?

Yes you can cast from the illusion

For the duration, you can cast spells as though you were in the illusion’s space, but you must use your own senses.

As long as the illusion is not also inside the cage, then casting the spell as if you were in that space is not an issue. By some unspecified means, the spell is cast through the illusion and not by you. So there is no spell attempting to pass through the barrier at all.

Just be sure that any sensory (eg being able to see the target) requirements of the spell can be met by the corporeal you from where you are.

And Invoke Duplicity itself is not a spell at all, so is not affected by forcecage at all.

Yes, you can cast spells through your illusion.

This question is really about 2 things: does Forcecage block Channel Divinity, and does it block the magical connection between you and your illusion?

Forcecage is very specific about what it blocks: matter, spells, teleportation, and ethereal travel. So things like sound, vision, and temperature still pass through the barriers of the cage, because we have no reason to assume they would be blocked. So because Channel Divinity is not a spell (rather, an action that you take to channel divine energy), it can be used through the Forcecage.

Similarly, Forcecage states nothing about blocking the connection between a magic user and an ongoing spell effect. For example, you would also be able to maintain concentration on a spell through the barrier of a Forcecage. And looking at the text of Invoke Duplicity, there is no reason to assume it would be blocked either:

For the duration, you can cast spells as though you were in the illusion’s space, but you must use your own senses.

So as long as you can see the place that you want to target with your spell while you're inside the Forcecage, you can cast spells through your illusion standing outside of it.