You can hack your boyfriend's Instagram using software that is downloaded onto his phone.
There are a few different kinds of hacking software that will give you access to his Instagram account:
A keylogger
I won't go into too much technical detail about how they work, but here's a quick explainer video:
The other option is to hack the Instagram app itself.
Many people claim to be able to do this, but very few are genuine.
Here's a pretty good example someone else posted on here a few weeks ago:
//technology.blurtit.com/3942448/how-to-view-private-instagram-photos
Before you go ahead and do any hacking though, I would urge you to consider the consequences.
I even found a news story that was about someone that went to jail for hacking into people's computers and web accounts:
"College Student Sentenced To 18 Months In Miss Teen USA Hacking Case"
Overall, I would say that whatever problem you are trying to solve between you are your boyfriend, hacking doesn't seem to be a solution at all.
It will just lead to more hurt and mistrust, and you could potentially be breaking the law too!
Instead, I would recommend having a conversation with your boyfriend about the issues you feel need discussing.
Openness, empathy and trust will always work better than spying and hacking!