@Adah N. it's definitely can be a difficult situation when doing large rent increases. I've only dealt with properties that are in rough shape, so it makes it a little easier when having to ask people to leave. The buildings had some problem tenants (one had drug dealers), and they were the easiest to deal with. I paid them to leave and they were gone in weeks. They were happy to get paid and I was happy to have them gone. This created vacant units that I could work on almost immediately. The other tenants I explained as empathetically as I could the units were in very bad shape and presented health/safety issues. Problems like rats in subfloor, mold everywhere in bathroom, rotted subfloors that you could stick your foot through with enough pressure, etc. The units would need to be made uninhabitable for weeks if not longer to bring up to standard. The old owner had allowed these issues to occur and they now needed to be fixed. I said they can take a few months to find alternative housing , offered to pay for some of the moving costs, and I'd fixed any immediate safety issues.
The bottom line is that units were in terrible shape, some had code violations, furnaces giving off excess carbon monoxide, and that these issues could not be addressed with them living in the units. I tried doing renovations with one tenant in the unit. Their unit wasn't that bad, but needed some plumbing work. Needless to say they complained the whole time, even though they specifically requested the work to be done. So I would not recommend it.
This approach won't work for everyone and frankly your situation seems different than mine. If the units are in decent shape, they might not take the increase as well. They will see it as price gauging and unnecessary. I agree with Nathan G. that doing gradual increase might just insult them more. You might want to just give them notice about the large increase. I'd personally give them at least 3 months to find other housing, but that's up to you.