Why did Skynet use nuclear weapons against humans, and not biological weapons? It is an AI. Nuclear blasts are as lethal to machines (due to EMP and radiation) as to humans. Also, they destroyed a vast amount of infrastructure, thus making Skynet weaker.
I can think of two good reasons:
Skynet had access to the US nuclear deterrent/first-strike system, and was able to activate and fire it. It wouldn't have had access to any viruses or biological weapons, as those aren't hooked up to any comparable systems. It would have had to send a Terminator to wherever those things are kept, steal them, and then manually deploy them. All within Skynet's capability, of course, but the issue then would be:
A virus takes time to spread. As of the time of writing this answer, we're eight months into the worst pandemic in a century, and it's "only" infected about 0.25% of the world's population. It took three of those months for it to truly begin spreading worldwide. We would, by that point in time, almost certainly have realised that Skynet was responsible for the virus' release and shut it down, one way or another.
In contrast, an MAD nuclear exchange is - by design - extremely swift and decisive. In Rise of the Machines we see it happen, and it's all over in a matter of minutes. There's no chance for anyone to prevent it, or take shelter, or attack Skynet in revenge, because by the time anyone's even realised what's happening, let alone that Skynet is responsible, it's already too late.
Not enough of them...
Biological weapons are strictly controlled by treaty and most countries maintain only enough samples to work on treatments and other defensive measures. If you turned loose all of the bio-weapon samples maintained in the world, you wouldn't have enough for wide enough spread infection to bring humanity to the brink.
...they're not weaponized...
Further, the bilogical weapons in the hands of the US and the USSR (now Russia) were maintained in labs, not inside active weapon systems ready for rapid deployment. Because of their fragile nature, biological weapons cannot just be left inside a warhead for years on end like a nuclear warhead can. It's also doubtful that many biological weapons would survive delivery by kinetic delivery systems of the type Skynet was put in charge of.
...and they're too unreliable...
Another problem with bio weapons is that their spread can be stopped fairly easily by a determined defender. In addition, countermeasures (vaccines and treatments) are already available for many bio weapons, and their lethality is never 100% and often a surprisingly lower.
...and WAY too slow.
When Skynet started the war, it did not have terminators and other such death machines in place and ready to go. Skynet needed time to build up its physical strength to destroy humanity. In order to do that, it needed to knock humanity for such an incredible loop that it would be impossible for humans to prevent the cyber war machine from getting to its feet. We could have fought a plague and prevented Skynet from comandeering weapon production facilities at the same time. But with everyone who knew better nuked and the population drastically reduced overnight, Skynet had the breathing room it needed.