I don’t know what to call the behavior of those who don’t believe that anybody acts with good intentions, so I'm looking for a suitable word, idiom or expression.
A skeptical person would usually mistrust anything, anybody:
a personal disposition toward doubt or incredulity of facts, persons, or institutions.
I don't like your strong skepticism about politicians.
The doctors still said they expected him to wake up any day, but Carmen was skeptical about how much they believed it.
For an adjective, try the British English term distrustful (or its British and American variant, mistrustful):
Feeling or showing distrust of someone or something:
I have grown up to be distrustful of men
Licinius became increasingly distrustful of him and suspicious of his own Christian subjects, whom he began to persecute.
It is not in borrowers' interests for them to become worried and distrustful of their lender.
Most Scots, they say, are distrustful of the nuclear industry and would prefer to tackle climate change through windmills, tidal and solar power, and recycling.
The nouns from which the adjectival forms are derived are distrust and mistrust.
Another possible adjective is the one you yourself mentioned in the question title, namely suspicious:
1 Having or showing a cautious distrust of someone or something:
he was suspicious of her motives
she gave him a suspicious look
No wonder people are suspicious of politicians and the political process.
In fact he was suspicious of any technological advance that might weaken the ‘master race’.
She is suspicious of doctors and nurses and takes her medication only episodically.
The noun from which the adjectival form is derived is suspicion.
(All definitions and examples from Oxforddictionaries.com.)