1. Remove density and height restrictions (e.g. "Berliner Traufhoehe").
2. Remove unnecessarily strict building requirements (e.g. "Daemmungswahn").
3. Remove legislation that is too tenant-friendly (e.g. "Mietbremse").
All three serve as breaks to increasing housing supply, the first is a legal impediment, the other two reduce investments.
Increasing supply outside of them, which would be easy because Berlin is surrounded by land you could build on won't change that. At least not without billions of investments in public transport infrastructure, maybe.
Any regulation like building requirements is probably federal, I doubt Berlin has any extra regulation. So there is nothing Berlin can do here.
Removing regulation that is too tenant-friendly isn't popular with voters, it's not going to increase the supply of apartments for students and poor people. This kind of regulation including but not limited to the Mietpreisbremse is also federal law and not something Berlin can do all that much about.
Even central Berlin is full of brown-field sites that could be developed -- Flughafen Tempelhof most prominently. If you go for Hong Kong style density there, a lot of housing problems would be solved ... Laws can easily be changed if there is political will at the local, regional or federal levels.
However, all this requires a bit of long-term thinking and planning, of which there is little evidence. Instead we get populist measures like "Mietpreisbremse" which hinder new developments.
Get rid of the restrictions on height, and build 50 - 100 floors Hong Kong style, at least in the center.
Tempelhof shows the stupidity of the Berlin voter. The refusal to develop even a small part of the airfield was almost exclusively justified with fears of "gentrification". Congratulation voters, you have just voted yourself into higher rents. (Also, one could have a new "Volksbefragung", in the light of the dire need for more housing in Berlin.)