Not unlike is correct. It means "like" or "somewhat like". It does not mean the same thing as "unlike".
Sometimes we say "not dissimilar to" as well.
Whether it is a double negative depends how you define that term. The type of double negative that is condemned by traditional grammarians and considered unacceptable in standard English is the type where a double negative is used to express a negative sense.
By contrast, the type of double negative where the two negatives cancel each other out and produce a positive meaning is standard English and completely acceptable.
Often the doubly negated meaning differs slightly from the positive one. For example, if I say "I was not displeased", that doesn't necessarily mean I was pleased. It could mean that I was neither pleased nor displeased.
As for your specific sentence that you're trying to write, you aren't trying to cancel out the meaning of "unlike", so you just want "unlike" (rather than "not unlike"). While "unlike with" is just about acceptable, it is better to use "unlike" on its own where this adequately conveys the sense: "Unlike the present perfect tense, the past simple tense employs certain time markers".
Of course, the simple past doesn't always involve a time marker, and the present perfect sometimes does use a time marker ("I haven't heard from him today"), but that's another matter.