Lukaku joined the Toffees on a season-long deal on transfer deadline day in what many regarded as a surprise move.
The Belgium international scored 17 goals for West Brom in the Barclays Premier League while on loan with them in 2012-13 and it had been widely expected that he would go on to feature prominently for Chelsea, who he signed for in August 2011, in this campaign.
But the conversations he had with the London outfit's manager Jose Mourinho about his first-team prospects convinced Lukaku that he would be best off - particularly with a view to next summer's World Cup - going out on loan again.
And asked if he felt he had a point to prove, the 20-year-old said: "Yes, because I think after last year, I deserve a chance - like I did the year before, before I went to West Brom.
"I thought I deserved that chance, I went to West Brom on loan and did well.
"This year, I have a point to prove first to myself, that I can do it again, and then to Chelsea, to show that I can be number one over there as well."
On his discussions with Mourinho, Lukaku said: "I had a few talks with the manager about playing regularly and the importance of it, because I couldn't afford another season on the bench or maybe playing one time out of three, at my age and especially with the World Cup coming up.
"So I spoke with him a few times to see his vision and at a certain point I thought going on loan again was the best thing to do.
"It was my choice. I wanted to play as much as possible.
"There is a lot of competition (at Chelsea), and I'm not scared of it, but there is a World Cup coming.
"I don't want people to just remember the Romelu Lukaku of West Brom and then for me to be in the background again.
"I want to kick on and push on after what I did last year, and improve in all aspects."
There had been speculation that Lukaku was set to rejoin West Brom on loan.
In signing for Everton, the frontman feels he has made a step up from Albion.
He said: "I thought this was the last step I needed. I felt the best solution was to come to Everton.
"They are a good footballing team and I felt a lot of confidence from the manager (Roberto Martinez). It is also that little step bigger than West Brom, I think.
"At West Brom I could show myself and I had the opportunity to play, and now I can do it, but for a bigger club."
Asked if he still saw a future for himself at Chelsea, Lukaku said: "I don't know.
"I think if I can do similar to last year, but for a club like Everton and with better statistics, why not? But we'll see at the end of the season."
When then asked if he was thinking he could possibly have a long-term future at Everton instead, Lukaku said: "Yes, you could say that as well.
"It was always my main ambition to go back to Chelsea and be a better player, but the end of the season is when you can have a balanced view of how things have gone and make a final decision."
Lukaku, who was at Anderlecht prior to Chelsea, is determined that he will not go down as a one season-wonder in English football.
"You always need confirmation after a good year so that people will take you seriously," he said.
"After only one good year, if you then have a down year the next year, people never take you seriously.
"I have personal targets that I want to keep personal. But I want to become a better player - a much better player, not just the same player as people know from last year.
"I hope that at the end of this season people say 'Romelu Lukaku is one of the best in this league'.
"I also want to help the team to finish as high as possible in the league, and hopefully reach a cup final."
Lukaku is raring to go, having been denied the chance to make his Everton debut on Saturday as they were playing Chelsea, making him ineligible.
Regarding the Premier League game at Goodison Park, which the Toffees won 1-0, he said: "I was thinking 'why am I not on the pitch at the moment?'
"It was difficult - every time I was thinking I would have done this or that.
"It was 90 minutes of happiness to see how the team were playing and also to see the Chelsea team, but also 90 minutes of pain because I wasn't on the pitch."
Everton's next match is Saturday's league trip to West Ham.