I want to indicate that I need help, with something particular, and that person that will help me, will help me a lot. Both variants in title seem incorrect to me, and I don't know how to write this correctly. Maybe using some punctuation?
"Help" is not a verb that goes well with the phrase "a lot" (used as an adverb). So you're right to feel that either of your examples are incorrect.
The best way to express your sentiment is to use "help" as a noun:
"I need a lot of help with this". "Please give me lots of help".
Edit: as suggested from the below comments, it's worth pointing out that you can use "a lot" when describing whether something has helped. For example, "Painkillers helped a lot" is indeed completely correct. However the difference is in the imperative/non-imperative form of "help". When talking about whether something helped, yes you can modify it with "a lot". But when you're either directing a person to provide assistance (or referring to such action), it normally needs to be unqualified.
"Help me!" can't really be modified, unless you add now/please - which is still the imperative.
Both could be used without confusion but the second would be preferred. In the first usage one could be using "a lot" to modify what they are doing and not the help they are providing.
It is misunderstanding seen in another question where sentences like "I was eating a sandwich in my shorts" is mistaken to mean the sandwich was in my shorts, not me (yum).