Sometimes people try to bring down a competitor slamming their site with tons of krapplinks anchored with dirty, adult keywords or other garbage anchors. But I'd try the opposite tactic: I'd go to SEMRush and see what Top 3-5 highest-traffic money keywords they were already ranking for -- and target those. I'd use MarketSamurai's Competition module to see whether those keywords were already over or under-weight in their backlink profile - and then hit them hard _with exactly what they want to rank for_.
The way Google is behaving post-Pengiuin if you link heavily saying 'my site is about keyword X/Y/Z' -- Google will put a -50 or -100+ penalty on THOSE X/Y/Z keywords and the organic traffic for those exact terms will utterly collapse. Suddenly, no keywords that really matter are on Page 1-4. They'll only be left limping on only the most obscure long-tail keywords for months or YEARS to come.
I know for a fact this works -- I've inadvertently did it to MYSELF on a few sites by pushing certain important keywords too hard!
P.S. I'd also try to get bad links placed on as many DEINDEXED root domains as possible. They're just utterly TOXIC these days.