[verb (plain)] + ふりをする [i-adj] + ふりをする [na-adj] + な + ふりをする [noun] + の + ふりをする
ふりをする means "pretend": someone acts as if something were true when it isn't. ふり is a noun, so it takes what any noun takes in front of it – a plain verb or an い-adjective directly, a な-adjective with な, a noun with の.
To fake a state, the verb goes in the た-form or the ている-form: ねた寝たふり is "pretending to be asleep", しんだふり死んだふり is "playing dead". To fake not knowing, hearing or seeing, it goes in the ない-form.
These go beyond the basic "pretend".
In the て-form, ふりをして links to what the person does under cover of the pretence.
In casual speech を often drops, leaving ふりする.