Introduction
Silver Samurai has bad-to-ok combos, until you put some meter in, in which his combo damage goes way up. Do note that Raimeiken can be mashed for more hits.
Silver Samurai can only do 2 ground normals or 2 air normals in a chain, unless he is in lightning mode, in which he can do a full magic series. Typically, you want to start with Cr. 
, as that is your low hitting move.
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
Normal Mode
D. Cr. > St. 
XX 
+ XX 
,
D. Cr. > Cr. XX 
+ 
Lightning Mode
All combos are in lightning mode (
+
), unless a switch is noted.
D. Cr. > Cr. > St. XX + XX ,
D. Cr. > Cr. > Cr. 
XX +
Ice Mode
All combos are in ice mode (+), unless a switch is noted.
D. Cr. > St. 
XX +
D. Cr. > Cr. XX + (aim up)
Fire Mode
All combos are in fire mode (+), unless a switch is noted.
D. Cr. > St. > XX +
D. Cr. > St. XX + XX ,
Ground to Air Combo
Normal, Ice, and Fire Mode
D. Cr. > Cr. > Super Jump > SJ. > SJ. > SJ. > SJ. > SJ.
Lightning Mode
D. Cr. > Cr. > Cr. > Super Jump > SJ. > SJ. > SJ. > SJ. > SJ.
D. Cr. > Cr. > Cr. > Super Jump > SJ. > SJ. > SJ. XX +
Corner Combos
Normal Mode
D. Cr. > St. XX + (mash) XX ,
Lightning Mode
All combos are in lightning mode (+), unless a switch is noted.
D. Cr. > St. > St. XX + (mash) XX , XX +
D. Cr. > St. > St. > Cr. > Super Jump > SJ. > SJ. > SJ. XX +
Ice Mode
All combos are in ice mode (+), unless a switch is noted.
D. Cr. > St. XX + XX ,
D. Cr. > St. XX +
+ , + (switch to fire mode), D. Cr. > St. XX + , +
Fire Mode
All combos are in fire mode (+), unless a switch is noted.
D. Cr. > St. XX + (mash) XX , XX +
D. Cr. > St. XX + XX + , +
Silver Samurai Infinite
This works in any mode. Remember that you can dash while pressing down. Dash just before pressing the LK.
Corner (D. St. > St. , crouch cancel) x n
Other Tips
Find an assist that allows you to combo off instant overhead J.. This is a very powerful move, and when combined with assists like T.Bonne projectile, Cyclops anti-air, or Thanos bubble, Silver Samurai can get a full combo. Synergy is key.