As an outsider, I don't think the Hawks are widely hated in the way the Patriots and Cowboys are, and not even close. The Cowboys are kind of a funny one in that nationally they're both the most popular and one of the least popular teams.
For fans that do hate the Seahawks, I suspect it's because they think they're loudmouths who play dirty and act like blowhards.
For a point of comparison, even as fans of NFC teams we all remember and know that Greg Schiano is a POS for his guys potentially injuring their opponents when trying to blow up a victory formation. That happened in 2012.
The Hawks have done that four times in the last four years, all against high profile opponents and IIRC all in nationally televised games (one of them being the Super Bowl). You do crap like that and you're gonna get the rep among unaffiliated fans for being scumbags.
TBF I think their rep even predates them too, as college football fans across country had given Pete the "Cheat Carroll" moniker looong before he even went to the Hawks, so when he went to Seattle, and with all the sideline fights and unsportsmanlike stuff, and PEDs, and practice violations and so on a fair number of people who are also college fans saw it and just said "yep, more of the same."
All in all I think "hate" is too strong of a word. You know how nationally the Bengals (deserved or not) have the reputation for being a bunch of criminals? The Hawks nationally (deserved or not) have the reputation for being a bunch of scumbags.
But again, I don't think they're hated or despised in the way the Pats or Cowboys are. People just think they're scumbags, which is kinda funny, given their QB.