Can someone tell me what ように means in this phrase?
あの人にそんな秘密があるようには全く見えないがな。
ように見える
can often be translated as "looks like". In your case,
あの人にそんな秘密があるようには全く見えないがな
He really didn't look like somebody, who was keeping such a secret.
This よう would be this one, written 様 and meaning amongst others "look, appearance".
There are lots of questions about よう(に). Reading some of them might be helpful as well.