Center Controller experienced a loss of separation event during a complex traffic period with multiple aircraft being routed through the airspace.

Narrative

Was quite busy at ZZZ Center. A lot of weather in and around sector. Seemed as though my airspace was the only route East and West of the weather. Many aircraft; not normally in sector; deviated into sector from other sectors/centers causing a greater than normal workload for myself and the sectors above and below me. Had been on position for well over an hour and not feeling well either. However; my illness did not factor into the error. I had passed a few break opportunities only because the workload did not permit me to give a solid briefing. In essence; I was too busy and my possible replacements did not appear to actually want to work the sector. As I stated; I do not believe any traffic in my sector at the time of the incident would actually be in my airspace on a normal day. Aircraft X on a weather reroute that took the aircraft into the weather the further West the aircraft progressed. I had the hand off on Aircraft X when he checked onto my frequency level at FL300 with traffic at 12 o'clock and 10 miles eastbound at FL310. Aircraft X seemed all to anxious to start deviating to the SW; but I could not do anything until I got him to FL340. I needed to do some coordination; traffic calls; and read some weather but I did a quick scan to climb Aircraft X because he was begging for it. After scanning I climbed Aircraft X to FL340; failing to see another aircraft Eastbound at FL330 about 40 miles West. This may be due to data block overlap of just failing to see the traffic. I jumped on the line to coordinate a lander at FL330 because I could not get him any lower due to deviating traffic at FL320 and FL300. As I was talking I did a traffic scan and saw the conflict prior to the Conflict Alert going off. I immediately jumped off the line and turned Aircraft X 25 right for traffic. I could not stop the aircraft because he was already out of FL322 at the time. I then turned Aircraft Y 25 right as well; but I doubt he turned at all due to weather off his right. I did not have a J-Ball in use because I did not have time. When I measured the traffic I had 4.7 miles. After the aircraft passed I went back and did coordination and other tasks and put both aircraft back on course.