I think OP's confusion arises from the proximity of the two participles, "kept" and "hidden".
I have a secret that I have kept hidden from my circle of friends and family.
comes from combining these two sentences:
I have a secret. + I have kept the secret hidden from my circle of friends and family.
The structure "that I have kept hidden..." is therefore a relative clause where "that" refers back to "a secret". "have kept" is a verb in the present perfect tense and in the active form. "hidden" is a past participle that does not form part of the verb phrase "have kept" but is used as an adjective that refers to the object "secret". We call "hidden" an object complement. Proof that "hidden" is not verbal (passive) but adjectival is that we can replace it with an adjective:
I have a secret that I have kept confidential.