heres how it works (at a highschool science level of info/interpretation):
all animals are in a Kingdom called Animalia.
This Kingdom is split into different Phyla (plural) one of which is Athropoda (which refers to jointed legs)
The Phylum Arthropoda is split into different Classes, one of which contains Insects (other classes of arthropods being those of spiders, scorpions and crabs.. all jointed legs see?)...
The Class Insecta is then divided into Orders, one of which is Orthoptera containing crickets (other orders include Lepidoptera- butterflies and Coleoptera-beetles)... and as an aside, the P in these names is silent, meaning Orthoptera is pronounced Ortho tera...
anyway, the Order Orthoptera is then split into Families...
and then Families are further split to a Genus level... and the Genus will contain individual species within it..
the Binomial name is the Genus of an organism followed by it's specific name...
so to finally answer your question, the cricket is in Phylum Arthropoda!