Providing your existing data in the database is CORRECTLY encoded in latin1, converting the tables to utf8 (using ALTER TABLE, as described in the docs) should just work.
Then all your application needs to do is continue doing whatever it did before. If your application wants to use unicode characters, it should set its connection encoding to utf8 and use utf8, but that's its own problem.
The problem is that a large number of crap web apps have historically sent utf8 data to mysql and told it to treat it as latin1. MySQL will honour this perfectly and save junk into the tables, as instructed.
Converting the tables from latin1 to utf8 will NOT repair this mistake, as you genuinely do have total rubbish in there. Repairing them is nontrivial, particularly if during the lifetime of the app it's been talking different types of rubbish to the database.