you become part of the empire when your capital province is inside the HRE. This can be done in 2 ways:
1. Move your capital to a province inside the HRE (the button for it is in the province screen).
2. Add your capital province to the HRE. To add a province, that province needs to border the HRE boundaries (which you say it does) and the empire needs to have a very high opinion of you (around 194 or something like that). Then, there's again a button on the province screen, the one with the double-headed eagle, which allows you to add the province, and you're in.
That's the first step. The next one is becoming emperor. To become emperor, you need to raise the opinion of several electorates to the degree that they prefer you over the current emperor. The peaceful way would be to use your diplomats to improve relations, to ally and royal marriage (if possible), to help them in wars etc. The more blunt, but faster way is to get into a war with them and force vassalise them. While this doesn't automatically lock-in their vote to you, you do get a +50 bonus for them being a vassal, and can now increase relations to +200 instead of +100, so in most cases you'd now be able to get them to vote for you quite easily. 2-3 electors are usually enough to become emperor.