וְשֶׂ֣רֶט לָנֶ֗פֶשׁ לֹ֤א תִתְּנוּ֙ בִּבְשַׂרְכֶ֔ם וּכְתֹ֣בֶת קַֽעֲקַ֔ע לֹ֥א תִתְּנ֖וּ בָּכֶ֑ם אֲנִ֖י יְהוָֽה
Grammatically speaking, "the dead" isn't even mentioned in the original Hebrew text. It was simply "the soul." In Hebraic thought, the soul is the unified body and spirit. The soul can be dead, or the soul can be alive. The text doesn't say one way or the other. But the word commonly rendered as "for the dead" is grammatically connected with the word commonly rendered as "cutting." These two words are שרט לנפש.
The word שרט /'seh-ret/ is simply some type of mark made by cutting. This could even be a tattoo, where a needle is used prick the skin to deposit ink. Strong even lists tattoo among the definitions for this word in his lexicon.
The word לנפש /law-'neh-fesh/ actually refers to a soul, alive or dead it doesn't say. The preposition ל is applied to this word; it could be any of: to, of, for, in accordance with, etc.
The remainder, לא תתנו בבשרכם /lo 'tit-nu bev-sar-'chem/, is as you would expect: put not on your flesh.
This might prohibit the mark of a(nother) person, such as a branding, or simply a mark to the person, even as in a self-inflicted mark.
The word כתבת /ch-'to-vet/ refers to something written.
The word קעקע /qaw-a-'qaw/ has a remarkably similar definition to שרט.
Again, Strong lists many definitions for this word, including tattoo. While we see examples of every other word here in other scriptural passages, this particular word only appears once in the scriptures. That leaves some questions, naturally.
And the remainder, לא תתנו בכם /lo 'tit-nu baw-'chem/, is a slightly altered version of the former, leaving out flesh: do not put on you.
Taking into account the above, we might have something more like the following:
And a mark to the soul do not put on your flesh. And a writing of a mark do not put on you. I am the LORD.
The second part is interesting, not only because the word for mark here does not appear elsewhere in the scriptures, but because of the process used to tattoo. Even today, a sketch (writing) is made on the skin, and then the skin is cut, scratched, or pricked, with a needle.
The preceding chapter begins specifically with the notion that what is to follow are prohibited because they are the doings of a pagan land. At the beginning of Leviticus 19, however, the LORD spoke to Moses telling him to address the children of Israel, that they shall be holy. The passage in verse 28 could be prohibiting something that is unholy, and not necessarily because of its pagan origin or connection. It could be an outright prohibition on markings on the flesh, or a specific type of marking common in that time.
Here are some interesting publications on the issue that focus specifically on the Hebrew words: