'cave lectorem tuus anima mea'
This is pretty close! Just some case and agreement issues.
Cave means "beware!" as a command to someone; caveat means "may [he/she] beware". So if you use cave you're talking to the person directly (cave canem "watch out for the dog!"), and if you use caveat you're talking about them (caveat emptor "the buyer should be wary").
If you want someone to avoid an action, the usual phrasing is cave ne plus a verb in the subjunctive. (Formally, this is called a "clause of fearing".) If you want them to fear a thing, the thing goes in the accusative. Here, it seems like you want neither: just a general "you should be afraid".
Lector ("reader") should be in the vocative if you're addressing them directly, or in the nominative if you're talking about them. Conveniently, the two forms look exactly the same for this word!
"Your soul" would be anima tua; "their soul" (if you decide to use caveat) would be anima ejus. The second word can be left off if it's clear from context.
"Will be mine" is literally mea erit; you might also use ad me veniet, "will come to me" = "will be mine". (I most likely got this phrasing from the Requiem mass: ad te omnis caro veniet = "to thee all flesh shall come".) If you want the subjunctive, that's veniat, with an a, or sit for erit.
All in all, my recommendation would be caveat lector ne anima [ejus] mea sit. "The reader should be wary, lest their soul be mine."