Introduction
Cyclops' combos are very damaging, especially when spending a bar of meter. He's also got some fairly easy combos. Any combo that ends in super can be mashed for more damage and more hits.
All combos can start with a J. 
+ 
for more damage, but you must dash before hitting the first button in the combo.
Explanation
">" denotes a chain.
"XX" denotes a cancel.
"," denotes a link.
"St." denotes a standing attack.
"Cr." denotes a crouching attack.
"D." denotes a dash. Dash before pressing the next button.
"SJ." denotes an attack while in Super Jump mode.
"J." denotes an attack while in Normal Jump mode.
Bread and Butter Combos
St. 
> St. 
> St. XX 
+ 
or 
+ 
XX +
Cr. > Cr. 
> St. XX + or + XX +
Cr. > Cr. > St. XX 
+ 
(mash for more hits) or XX + (first hit of Gene Splice, hold up)
Ground to Air Combo
D. St. > St. > St. > Super Jump > SJ. > SJ. > SJ. > SJ. > SJ. > SJ.
+ > Super Jump > SJ. > SJ. > SJ. > SJ. > SJ. > SJ.
If the j.HK is dropping, try slowing down the pace you're pressing the buttons at.
Corner Combos
D. St. > St. > St. > XX + or + XX +
D. St. > St. > St. > St. XX + (mash for more hits) or XX + (first hit of Gene Splice, hold up)
Throw, Cr. > St. XX + (mash for more hits) or XX + (first hit of Gene Splice, hold up)
Cyclops Infinite
Does not work on Xbox 360 or PS3 due to a bug in that port. Does not work on Sentinel.
+ or + > Super Jump > SJ. > SJ. > SJ. , (short delay) SJ. + > SJ. +, SJ. + > SJ. + > SJ. + land and relaunch.
All the combos that go from to + are links, so you're going to need to adjust your timing as needed. That's the basic "Easy" combo but he has several variations on that, all challenging in their own way.
This is not an easy combo by any means. Many Cyclops players never even try to go for this, so don't feel bad if you can't get it.