Ο λεκές δεν φεύγει εύκολα, γι’ αυτό βάζω λίγο απορρυπαντικό πάνω του.

ο is the masculine singular definite article (the).
So Ο λεκές = the stain. In Greek, common nouns normally need an article in everyday speech, and it also marks the noun’s gender and case.

λεκές is masculine. You can tell because:

• it takes the article ο (masc. nominative singular),
• many masculine nouns end in -ς (though not all nouns ending in -ς are masculine).

Greek has two common negators:

• δεν negates statements (indicative mood): δεν φεύγει = it doesn’t come off
• μη(ν) is used mainly with commands, requests, wishes, etc.: μη(ν) φύγεις! = don’t leave / don’t go!

Here it’s a factual statement, so δεν is correct.

The verb φεύγω literally means leave/go away. With things like stains, stickers, dirt, etc., it commonly means come off / go away / lift out.
So Ο λεκές δεν φεύγει = The stain won’t come off / doesn’t come out.

Greek present tense often covers:

• general/habitual meaning: doesn’t come off easily
• what I do now / in this situation: so I put a bit of detergent on it

Depending on context, English may choose present, progressive, or (near-)future, but Greek can keep the present.

εύκολα is an adverb meaning easily.
It modifies the verb: δεν φεύγει εύκολα = doesn’t come off easily.

(Contrast: εύκολος / εύκολη / εύκολο are the adjective forms = easy.)

γι’ αυτό means that’s why / for that reason.
It comes from για αυτό, but the α in για often drops before a vowel, so it’s written with an apostrophe: γι’.

So: ..., γι’ αυτό ... = ..., so/that’s why ...

Because it’s linking two independent parts: 1) Ο λεκές δεν φεύγει εύκολα
2) γι’ αυτό βάζω... (= that’s why I put...)

In Greek it’s very normal to separate these with a comma, similar to English ..., so ...

βάζω is 1st person singular present: I put / I’m putting.
Greek doesn’t need an explicit subject pronoun (εγώ) because the verb ending already shows who does it.

λίγο means a little and here it’s neuter singular to match απορρυπαντικό (also neuter singular).
So λίγο απορρυπαντικό = a little detergent.

(You may also hear λίγο used more generally like English a bit, but in this phrase it neatly matches the noun.)

It’s the direct object of βάζω (I put), so it’s in the accusative.
For neuter singular like απορρυπαντικό, nominative and accusative look the same, so you don’t see a change in form.

πάνω = on / on top.
πάνω του = on it (literally on top of it).

του is the genitive clitic pronoun (of it / of him) used after πάνω in this common construction. Here it refers back to ο λεκές (masculine), so you get του.

You could also say πάνω σε αυτόν (on it/on him), but πάνω του is very natural and compact.