Depends what bioweapon you want
One of my degrees was in biological sciences & did some post graduate research work in a biochemistry lab before career switch.
Unless the quality of course content & students has dropped massively there will be an awful lot of graduates out there able to create bio weapons.
It really depends what type & amount of weapon you want as some things would depend on expertise and specialist equipment that should set alarm klaxons going, others could easily be achieved with relatively basic equipment (and skills).
Main hassles for the person creating it would be
a) Avoiding exposing yourself in the processes (it's not just creating the bioweapon of choice its refining it afterwards..).
b) A suitable delivery system for the weapon... As a decent delivery system is going way outside biosciences skill set & into the engineering realm (unless you go the suicide* approach of human delivery)
* Depends on the weapon, plenty of things you can be vaccinated against that would cause disruption (after all huge death tolls may not be the aim, casing a lot of chaos & the occasional casualty may be the aim) in a "Western" country as most of population not vaccinated against it e.g. typhoid.