You can't become an Arcane Trickster unless you cast Arcane spells. Although a spell may be arcane, a Cleric with Anyspell casts it as a Divine spell. For example, if you use Anyspell to cast Burning Hands, do you have to deal with Arcane Spell Failure? Of course not, because you're casting it as a Divine Spell.
Even if you could gain access to the class, it would provide you with little or no benefit to your spellcasting ability. If I'm not mistaken, Arcane Trickster only advances your Arcane Spellcaster level, and if you manage to take it as a Cleric, you don't have an Arcane Spellcaster level. So you wouldn't have a class to add it to. So you'd be getting the special abilities, but your Cleric spells would be exactly the same.
Even if you DID have levels of an arcane spellcasting class (such as one level of Bard or a level or two of Assassin, unless you qualified for Arcane Trickster with that class (thus making this entire issue pointless), you couldn't add onto it. Just like if you have 13 levels of Wizard and 1 level of Sorcerer, you can't have the Archmage class add onto your Sorcerer ability.