I've been doing some testing and I think I've come up with the most efficient (albeit the most tedious!) way to increase your Hero Level quickly.
Load up the Character specific Harm Room for your chosen character on Challenge I and go through the first few challenges (perfect evades, parrys etc)
On challenge 3 or 4 you will have to use your support ability (LB) to defeat enemies ... Don't!
Keep spamming power attacks to beat them and infinite waves will continuously spawn in and your XP will increase in no time! I did this for an hour and went from Level 20 to Level 28. So roughly 8 Levels an hour or 1 every 7 and a half minutes.
I took a break, assigned my skill points and got another 8 levels in an hour (so it appears XP needed doesn't increase per level, or very slightly if so).
You'll need to get your parrying down as even in Challenge Level I you can die pretty quickly, but the game loads you immediately back to that checkpoint to continue.
PLEASE NOT THIS IS SIMPLY FOR HERO LEVELS. YOU WILL NOT GET GEAR VIA THIS METHOD TO INCREASE YOUR POWEL LEVEL.
I compared this to running Drop Zone missions and in the same time I levelled up 5 times, this was simply mainlining the objective, not searching for resources and playing Solo so as not to waste time waiting for Matchmaking.
I tested this with Cap but this may be even quicker with characters such as Black Widow or Thor who can overcharge to deal more damage and hence kill more enemies quicker.
I also can't say for certain but this MIGHT in the long run cut down time on the Power Level grind as missions should technically be easier and therefore quicker if you go in with all skills unlocked.
Hopefully this helps! Feel free to message me if you have any queries.