How strong is this set theory?

Even if $φ$ is restricted to not use $C$, the theory is inconsistent! Here is the simple 2-line proof.

Let $R$ be such that $∀x\ ( x∈R ⇔ C(x) ∧ x∉x )$ by Comprehension.

Then $C(R)$ by Construction. Thus $R∈R ⇔ R∉R$. Contradiction.

Suppose first that $\phi$ in the comprehension schema is allowed to use the symbol $C$. Then I claim your theory is inconsistent. Indeed, suppose we had a model $M$ of this theory, and let $M'$ be the submodel consisting of the elements of $M$ that satisfy the predicate $C$ in $M$. Then every definable-in-$M'$ family of elements of $M'$ is also definable in $M$ (by a formula that may need $C$ to restrict quantifiers) and therefore, according to comprehension, given by an element $x$ of $M$. By the axiom of construction, $x$ satisfies $C$ in $M$ and is therefore in $M'$. Thus $M'$ satisfies unrestricted comprehension, which is impossible by Russell's paradox.

Now suppose the $\phi$ in comprehension is not allowed to use $C$, so it's a formula in the language of ZF. That causes a problem in the preceding paragraph, because a definable-in-$M'$ family of elements of $M'$ might not be definable in $M$ without using $C$. Specifically, if the definition in $M'$ involved a quantifier, the corresponding definition in $M$ would need to refer to $C$ in order to make the quantification range only over $M'$ and not all of $M$. Fortunately, that doesn't matter, simply because the formula involved in Russell's paradox doesn't contain any quantifiers.

So even with the weaker, $C$-less-$\phi$ version of comprehension, the theory is inconsistent.

Regarding the latter suggestion to restrict $\phi$ to positive formulas. The resulting theory would be consistent relative to positive set theory, but it would be just a redundant re-exposition of a fragment of it. Simply take $C$ to be the predicate of being "equal to itself" i.e.:

$C(x) \leftrightarrow x=x$,

and comprehension would just be positive comprehension, and axiom of construction would be trivially true! Same would apply if for example we've restricted $\phi$ to stratified formulas, we'd only get Quine's New foundations set theory.

Actually that argument would hold for any kind of a theory that has a naive like comprehension axiom with only syntactical restrictions on the defining formula.

I think the only reasonable adjustments to salvage your approach are those you've had with your original theory, the one you raised this theory in connection with, which I think is as strong as $PA$, if you want to strengthen it, you add an axiom of infinity, or even a more risky approach is to add a universal class of all $C$ objects but with changing the bound on parameters in comprehension (the schema of construction) for all of them to be elements of that universal class. But I don't know if those are consistent, and what would be their consistency strength?