No, the Fiio E5 is very entry level that works well for the price but won't efficiently drive a 600ohm headphone.
You'll need a desktop amp. Fiio is actually about to release the E9, which is their desktop amp, and so far from initial impressions, it may be the new price/performance king, but don't quote me on that. As far as for portable use, I don't think Beyers are gonna do particularly well outside...
I'll paste this off the pm I sent you...
Okay, I see.
The typical audiophile prefers a balanced sound across the entire frequency. Two of the most popular offerings are from Beyerdynamics.
The DT880 600ohm (open backed headphone, so sound leaks out, but sounds airier, and has more width and depth to the sound)
and
DT770 600ohm (closed, giving you isolation and minimal sound leakage, as well as stronger bass)
are well loved here as great audiophile headphones in the $250 price range.
As far as hard to drive headphones go, these two aren't actually THAT hard to drive, so a somewhat decent amplifier can help drive them good enough, until you can afford a really good amp that will cost you nearly as much as your $350 budget. I don't know what amps work well for these two, but I have the DT990/600 and plan on getting the Fiio E9 which is gonna cost me $130 or less, to go with my Fiio E7 portable amp/dac. Another cheap offering that can tide you over until you have some money for a good amp is the Bravo V2 which can be found on ebay on auctions that go for normally $30-80, and is pretty powerful for the price. I have friends online that own it and love it, and they both own DT880s.