The Hammers have enjoyed a solid start to their return to the top flight with wins over Aston Villa, QPR and Swansea and draws against Norwich and Sunderland leaving them with 11 points from seven games.
The midfielder believes there is enough talent at Upton Park for them to have a good campaign in the Premier League, although he knows tough games coming up against the likes of Manchester United, champions Manchester City, Liverpool, Newcastle and Chelsea will test them.
O'Neil told the club's website: "We have had good results in the first few weeks and started to build the foundations of what we want to achieve.
"In the last week, we had the great win away at Queens Park Rangers and then against Arsenal, we came out and made a great start to the game, but ultimately came away disappointed after losing to what is a very good Arsenal side.
"But we have to look to build on what we have done well so far in the next few weeks again."
He added: "There is a really good vibe around the place. The lads have done fantastically so far this season. We have had a decent start and coming up from the Championship is hard, so getting a good beginning to the campaign is important.
"Yes, we have tough games coming up but the form we have been in and the squad we have, we are all looking forward to the challenges ahead and believe we are capable of meeting them."