If I have a Mexican passport but I'm a permanent US resident, do I have to go through immigratiin and customs when I land in Mexico?
Yes. Everyone has to go through customs and immigration controls in some form or other. Diplomats enjoy some exemptions, but unless they have a very high profile, they still have to present themselves at the immigration counter and show their diplomatic credentials to prove that they are entitled to those exemptions. Even if they are sufficiently important to bypass that part of the process, a staff member will typically present the passport to immigration officials for whatever formalities are required.
Similarly, a "regular" Mexican national has to show a Mexican passport or substituting document to an immigration officer as proof of Mexican nationality before the officer can give that person the rights and privileges of a Mexican national.
As far as customs goes, Mexican nationals are, like everyone else, subject to restrictions on the nature and quantity of goods they can bring into Mexico, so they are also subject to customs controls.
The fact that you reside in the US may affect the customs exemptions that apply to you, but that's more likely to reduce the exemption. Regardless, it doesn't change the fact that you need to clear the customs checkpoint.
If you are arriving by air, it's like any other country. You present your papers, assure them that you are who you say you are, and that you are not carrying contraband or items in excess of allowable amounts for tax-free entry, if not you pay tax. Same thing if you are driving across a land border.
Walking across a land border with the U.S., on the other hand, might be pretty slack on the Mexican side, you might not actually encounter any officials. Don't worry, it will be made up for (and then some) upon return to the U.S. side. (This can actually be an issue for foreign nationals because you are supposed to have a Tourist Card to go deeper into Mexico, and you could easily go to the bus station and get yourself into potential difficulty).