In order to target that player afterwards, you must use CommandStats to track the success of a command.

Prerequisites:

Objective to hold the return value.

/scoreboard objectives add AboveChest dummy

Applying the "AffectedBlocks" trigger to all players, who will then set their own "AboveChest" score depending on the success of block-related commands. This may need to run on a clock if new players can join at any time.

/stats entity @a set AffectedBlocks @a[c=1] AboveChest

In order for CommandStats to modify a target's score, that target must be tracked prior. This may also need to run on a clock.

/scoreboard players add @a AboveChest 0

Clock commands:

The following must be run on a clock in numerical order.

1. Cause players to run a /testforblock to find the chest. If the command is successful, that player will have their "AboveChest" score set to 1. If unsuccessful, that player will have their score set to 0.

   /execute @a ~ ~ ~ testforblock ~ ~-1 ~ minecraft:chest -1 {Items:[{id:"minecraft:stone",Count:64b,Slot:0b}]}
   

2. Target players based on their "AboveChest" score.

   /say @a[score_AboveChest_min=1] is above a chest containing a stack of 64 stone.
   

1.13+ Update to Skylinerw's answer

1.13 has completely changed commands. Are you playing 1.13 and have the same question? This is the answer for you.

Prerequisites

1. Objective:

   /scoreboard objectives add AboveChest dummy
   

2. Set the player's initial score.

   /scoreboard players set @a AboveChest 0
   

/stats is deprecated, use /execute store instead.

Clock commands

1. Find chest and store success/failure in scoreboard variable.

   /execute as @a at @s if block ~ ~ ~ chest{Items:[{id:"minecraft:stone",Count:64b,Slot:0b}]}
   

2. Target players who have a score of 1. Use whatever command you want and this target selector to get them

   @e[scores={AboveChest=1}]