Minecraft – Blockdata clearing chest instead of putting item in chest

A single item should be an unnamed compound tag and, for a chest, all items should be contained within the same list as each other called "Items". Compound tags use curly brackets: { }, and lists use square brackets: [ ].

Currently, your problem is that you have each item in its own list, when they should both be in the "Items" list.

You are also using numeric IDs which won't work in future versions.

The correct command should be:

/blockdata 3 56 0 {Items:[{id:command_block,Count:1,Slot:12,tag:{HideFlags:1,Unbreakable:1,display:{Name:"Command Block",Lore:["Write Commands Here!"]},ench:[{id:4,lvl:2}]}},{id:stone_button,Count:1,Slot:13,tag:{HideFlags:1,Unbreakable:1,display:{Name:"Button",Lore:["Activate Commands!"]},ench:[{id:4,lvl:2}]}}]}

Essentially, I've changed:

Items:[{command block item info}],[{button item info}]

To:

Items:[{command block item info},{button item info}]

Does this need to happen particularly quickly? Because if not, you could summon tnt or a fireball to destroy the cloned chest so it drops its items, use /entitydata to change their display names, then teleport them to a hopper above another chest. This is a comparatively slow workaround and leaves you with an extra chest (the cloned one that was destroyed will drop as an item), but you can't edit an item's data without completely overwriting it unless it's in entity form. Hope this helps!