I've lived in the area for twenty years in tenement flats. During that time the top end of Easter Road (near London Road), Hillside and the Abbeyhill colonies, have gentrified considerably. Hence shops like Cornelius, Manna House, nice cafes etc. It's still a mixed area, having been at one time solidly working class. Gets more 'authentic' Leith the further north you travel, though even that is changing as young bohemian 'hipster' types are priced out of the city centre/Southside rental markets and are gravitating to Leith and Easter Road. I've heard it said that Leith is the new Hackney! :-)
Good bus connections on London Road will take you almost anywhere in the city, and also out into East Lothian. Not far to walk (or cycle) to Waverley for trains to everywhere. Also very pleasant to walk (or cycle) into town via Regent Road. Plenty of shops nearby, convenience and small or specialist shops in Easter Road itself, big chain retail and Sainsbury's a short walk further east at Meadowbank shopping centre.
Tenement life is in general quiet and hassle free, top floor generally the most peaceful if you can get it. Ground floor flats will be noisier though if facing back green can be very appealing. If you are lucky your block may have a pleasant back green where you can spend time having a summer barbecue, gardening or just hanging out on a nice day.
Of course there's always the chance that a bampot will move in to a rented flat on the stair and throw wild parties/cause mayhem. In my experience these people generally don't last long as neighbours - if they don't self-destruct under their own steam, then strongly worded letters to letting agents from annoyed residents usually modify their behaviour or see them leave.
That said, tenement stairs can vary a lot. If there is a majority of rented flats, then the population may be very transient leading to a rather anonymous feel to the stair, strange faces constantly. On the other hand if your neighbours have been there for a long time you may find there are stair cleaning rotas, regular residents' meetings, and unwritten rules you need to discover.....locking your bike in the stair may or may not fall foul of one or more of these.
I find the area to be pretty safe and a good place to stay. As with anywhere in central Edinburgh (and the top of Easter Road is really quite central) it can get a bit 'exciting' at night on occasion. I have witnessed a few bits of 'bother' on the streets over the years but none of them have involved me. Very rarely caused by actual residents, more often folk from elsewhere who are visiting a local hostelry or attending a football match at the nearby Hibs ground. Again though, the further north you are, the more 'interesting' things tend to get. Personally I would recommend staying south of the Powderhall railway line, though there is a slightly posher bit of town facing Leith Links and extending up Lochend Road.
Most of the bars south of the railway bridge on Easter Road are fine to take a pint or nip in, even if you're not a regular. Further north, you need to be more careful in my experience: some bars have a distinct "regulars only" vibe (Iona Bar and Persevere notable exceptions here).
Oh yeah, if you own a car you may find it a constant struggle to park on street: but then that goes for most of central Edinburgh...