Because it can get extremely hot, most people recommend going early morning to beat the heat.The main issue with that is you are going to run into peak traffic if you go early morning. Huge amounts of traffic are travelling from the western side into Honolulu for work, school etc.
The car park at Diamond Head opens at 6am and quickly fills up. There is parking available outside the park but it adds another 15mins or so of walking.
Are you only staying in Aulani? Ie not doing any stay in Waikiki - some sites are just easier to negotiate from there, Diamond Head being one.