I'm looking for advice about completing the Highline in spring - as early as possible between April-June.
I assume you are skiing it? If so, Highway 150 doesn't open until late May or June, so it would be a long ski in. The road to Chepeta Lake opens sometime in June and sometimes July, so I would go west to east.
If you are on skis, I'd still take an ice axe for Deadhorse Pass until at least July. An ice axe can be useful for the east side of Rocky Sea Pass unless you are a really good skier.
April and May will have complete snow cover, but depending on snowfall, June can have bare spots, especially late in the month.
June can be a tricky month since before the middle of the month, most areas will have complete snow cover, while some areas can only have partial snow cover with lots of mud.
Before mid June, bring lots of fuel since all lakes and streams will still be frozen over.
Although snow persist until at least early July in most years (last year was really dry and warm) you should find running water from mid June onwards; earlier at the lower elevations.