Bill enlarged Nato considerably during his tenure, and it grew even more later, unnerving the Russians. Additionally, there was the intervention in Yugoslavia, which nearly resulted in a conflict between US and Russian troops over the airport. Hillary's foreign policy would have been a continuation of Bill's.
Ultimately, neither the US nor Russia actually wants WWIII. That's why the various incidents always "nearly resulted" rather than actually resulted. That's even more true now than it was during the Cold War, which also saw plenty of "nearly" conflicts. At the end of the day, both sides will always find some way to save face. We had an incident in 2015 where a Russian fighter was shot down by a NATO aircraft (which is more or less an explicit act of war, on both sides if you accept that it was in Turkish air space when it happened), and both sides made a lot of bluster but again, WWIII was avoided. At the end of the day, no one sane actually wants a war between the US and Russia, and Clinton is sane.
There's this image (mainly coming from the primaries and the memory of the Iraq vote) of Clinton as a giant war-monger. She isn't. Sure, she was a bit more hawkish than Obama, but not to an extraordinary extent. There's a reason he was willing to make her Secretary of State, and it's worth remembering that one of her first big initiatives was the "Reset Button" with Russia. That didn't pan out in the long term, but that had more to do with Putin's domestic needs than it did with Clinton. And of course, Bill Clinton got along famously well with Boris Yeltsin during his presidency (during which, again, WWIII failed to break out).
More broadly, there isn't really that much difference between Obama and Clinton. They are both normal, blue-state Democrats, and are constrained by similar pressures. Either one will face a Tea Party insurgency in 2010 and be essentially reliant on executive orders after that.