Hackney is like a lot of inner London, transformed in the past 20 years or so.
Hackney is the borough but it has a lot of neighbourhoods. London Fields is nice- with million plus homes around the park , Broadway Market feels like nowhere else in London, there are some great primary schools around there too. The area around Victoria Park, called by agents, "Victoria Park VIllage' is another area full of expensive homes , coffee shops and trendy restaurants, deli's and bookshops - to be honest neither of these areas are diverse as such, mostly white middle classed and uber trendy. It's a bit more mixed up in areas like Dalston, Haggerston and Clapton , I used to live there when I was younger for a while. (Dalston) Hoxton and Shoreditch are just party zones and full of bars and nightclubs and restaurants, very expensive property too. But great for a night out.
I think the area you'll like the most , the most family friendly by far is Stoke Newington - lots of middle classed families who have moved in from outside, but has retained a diverse feel, as social housing mixes with beautiful period housing stock.
Mossbourne is probably the best local secondary, the good primaries outnumber the secondaries as middle class parents tend to send their kids further afield either private like City / Hampstead schools or good state schools outside of Hackney over in Camden , Crouch End.
I know one family that lived near victoria park that sent their kids to Hammersmith schools, its do-able as Hackney is so central and easy to get anywhere.
Given the lack of choice for secondary, you may be better off moving to somewhere like Tooting which still feels like diverse London, but you have easy access to grammar schools down in Sutton , privates in South West and West London. Price wise its slightly cheaper, as Hackney property market is absolutely CRAZY expensive- it's the reason we sold up and moved south .Somewhere like Tooting has the culture of inner London, but you've access to good schools for your DC. Hackney for newcomers is still very much geared for childless trendies, or cool couples with small children in "Stokey"