Since Hellenism refers to the height of Greek culture, especially in regard to its influence and colonization, Keats is a Hellenist in the sense that he continues to spread Greek culture through his poetry.
Keats discusses the figures on the urn as pristine and perfect since they are frozen in time. They do recall a civilization at its zenith, so this perfectly preserved image - which is perfect because it is a frozen snapshot and cannot die (at least, the image of the icons cannot die) - represents the Hellenistic culture at its finest. However, Keats does come to the conclusion that the figures, perfect in their frozen image, are silent and they do not consummate their love and therefore, cannot die but do not live.
And in general, Keats is an aspiring hellenist because he contemplates what it would be like for those figures on the urn to be alive, or what it was like when they were.