This book attempts to explain the grammar of modern Japanese as clearly and systematically as possible.
Literal
This book [topic-は], modern Japanese [gen-の] grammar [obj-を] as-much-as-possible understandably, systematically explain-try-to-do book [copula-です].
できるだけ 'as much as possible' is a set adverb. 体系的に 'systematically' uses the productive suffix 的 to create an adverb. ~しようとする 'try to / attempt to' combines the volitional ~しよう with とする for deliberate effort. The sentence is itself a meta-description — a grammar book describing its own purpose.