Word (Verb?) for when a criminal (or soon to be criminal) escalates their behavior

The closest match to the meaning you suggested, that also fits with the sentence you supplied, is the verb

degenerate, v.
1. intr. To lose, or become deficient in, the qualities proper to the race or kind; to fall away from ancestral virtue or excellence; hence (more generally), to decline in character or qualities, become of a lower type.

["degenerate, v.". OED Online. June 2016. Oxford University Press. http://www.oed.com/view/Entry/49056 (accessed August 26, 2016). Emphasis in definition mine.]

In your sentence, 'degenerated' works well:

John used to just stare creepily at his classmate, Jane, all throughout English 101 but today his behavior degenerated and he gave in to the urge to follow her back to her dormitory after class.

Another word that works in your sentence, but doesn't seem to match the nuances of meaning you supplied as well as 'degenerated' does, is 'deteriorated':

deteriorate, v.
....
2. intr. To grow worse in character; to become lowered or impaired in quality or value; to degenerate.

["deteriorate, v.". OED Online. June 2016. Oxford University Press. http://www.oed.com/view/Entry/51216?rskey=m1OmcN&result=2&isAdvanced=false (accessed August 26, 2016).]

Notice that my second suggestion is defined in terms of the first. The first more strongly suggests volitional moral or criminal degradation ("the qualities proper to the race or kind") than the second.

Escalate — TFD

escalate into something

to intensify into something; to increase gradually into something.

"This argument is going to escalate into something serious very soon."
"These cases of the flu could escalate into a real epidemic."

escalate something into something

to cause something to intensify.

"He escalated the argument into a vicious fight."
"The dictator tried to escalate the disagreement into a cause for war."

See related meme: "That escalated quickly!"

He is taking his bad behavior (or criminality) to another level.

ODO

Further improve or develop something that is already successful

"five years after founding my software company, I’m ready to take it to the next level"

In some Common Law jurisdictions a petty criminal may be charged with any number of different offences which are collectively labeled a misdemeanor, but those criminals who commit more serious crimes are charged with offences that are collectively labeled a felony. Such a person may be described as a felon. An alternative would be the word recidivist, one who is a repeat offender and, in my opinion, covers the quantitative aspect of the OP but not necessarily the qualitative aspect of a person's burgeoning criminal behavior.

_{Not single words, but probably relevant idioms.}

Fat hit the fire — TFD

Fig. A situation that suddenly becomes frantic and unpleasant. (Typically: suddenly ~; then ~; when ~.)

"Things were looking bad in the stock market, then the fat hit the fire and I lost everything."

Go from bad to worse — TFD

To progress from a bad situation to one that is worse.

"Things went from bad to worse in a matter of days."
"I'm afraid that things are going from bad to worse."

Kick it up a notch — Online Slang Dictionary

verb phrase To increase the intensity of something.