If someone has interest (means that they are doing something to achieve that) in being an academic, such as a lecturer for a university, if that person is having educational qualifications which are suitable for that lecturer position and if that person has not become a lecturer yet and if his/her profession is something else now, what is the best word to call that person? Can he/she be called as academic enthusiast?
Someone who has set a goal and is actively working towards it is aspiring to (toward?) said goal.
as·pire (verb) direct one's hopes or ambitions toward achieving something.
"we never thought that we might aspire to those heights"
While used with a wide range of aspirations, a common use is applied to struggling actors or artists.
You can use that phrase, though I'd recommend you do that among friends and not on job materials. My answer will address commonly recognized titles within academia.
There is a term I've encountered at conferences for people who are not affiliated with a university but who do academic research: independent scholar. This substitutes for their affiliation, but it also serves as an ad hoc position title. They even have their own organizations, like the National Coalition of Independent Scholars.
While there is not necessarily a sense that independent scholars aspire to be university professors (some do, some don't), it would be preferable to academic enthusiast or aspiring academic in many job-related contexts. University departments are often looking to hire graduate students or post-docs as faculty. The title enthusiast sounds comparatively nonchalant and amateur compared to those positions, and aspiring sounds hopeful but insubstantial. (OK, so you aspire. What are you doing to get there?) At least in the US, Canada, and UK, hiring committees show a strong bias towards people who are currently doing the work they would be hired for in some way. This bias has led to phrases like "publish or perish" to describe the expectation of academic publishing to get or keep a job. Because hiring committees expect current competency and not hopefulness, I recommend avoiding aspirational language in any job materials or portfolios.
In comparison, independent scholar isn't a perfect term, but accompanied with a publication history it would demonstrate that the prospective applicant is serious about their work. Other terms that might be used include prospective faculty (used by several university websites) and faculty candidate (often applied internally by universities to applicants for faculty positions; also used generally as a label for people on the faculty job market). In other words, I recommend picking explanations that signal how you're one step away from the next position.