HCF Controller described a potential conflict event when they failed to closely observe the climb performance of one of the involved aircraft.

Narrative

Sector 5; KOA Runway 35 in use; a CRJ2 departed KOA airport headed west bound to HNL climbing to FL220. There were two aircraft following in sequence headed to HNL. I stopped the CRJ2 climb at FL200 so they could level off sooner and pick up speed. The adjacent sector had opposite direction traffic into KOA at FL190 with a pilot's discretion to 3;000. When I handed off the CRJ2; there was about 50 miles between the 2 aircraft and the CRJ2 was climbing out of FL180. Both I and the other Controller saw the traffic but assumed we would have vertical well before they would became an issue. Meanwhile; I had a A320 inbound to KOA on a 180 heading to keep him clear of my departure corridor off KOA. KOA Tower requested a release for an E135 and I assigned them a 350 heading. When I went back to switch the CRJ2 to Sector 4; I noticed his altitude was FL189 and had been there for about 1 minute without changing. It looked as though the CRJ2 had level off on his own. At this time the CA (Conflict Alert) went off and I told the CRJ2 to verify they were climbing and issued a heading left of course. The Sector 4 Controller turned his aircraft and instructed them to descend. I'd say the closest proximity was 7-8 miles horizontally. No separation was lost. When the Sector 4 Controller told the CRJ2 that it looked like he had leveled off at FL190; they said they had experienced light rime ice and had told myself that they would be in a slow climb. This so called conversation never took place and I was never aware of it. Even if it did; leveling off for 1 minute at the wrong altitude for direction of flight is not the same as a slow climb. While all this was going on; the E135 had departed and been RADAR contact. I noticed their 350 heading looked a little left of course and was a little closer to the A320 on the downwind. So I cleared the E135 direct EBBER which is directly to the east (about a 060 heading from the position) to get more spacing and get them on their way. The two aircraft were never in danger of each other; but they did look a little close for comfort. I'd say about 6 miles. Though it should have never been that close; I feel now that my attention was diverted to the situation with the CRJ2; and because of it; everything else was secondary. Recommendation for my part: make sure I watch the aircraft climb all the way to altitude. Also; make a conscious effort to scan the entire sector instead of just focusing only on the one matter. I feel pilots need to understand they cannot just do things on their own without advising ATC of what they are doing.