The arduino library supports A/D conversion at about 13 KHz, not fast enought for a useful audio-rate oscilloscope. Rewriting the acqusition loop to directly access the ADC reqisters speeds up conversion (including loop overhead) to about 688,000 samples/sec, a considerable improvement. The sampling loop becomes quite short. If we define registers and constants as follows: 

#define ADC_MR * (volatile unsigned int *) (0x400C0004) /*adc mode word*/
#define ADC_CR * (volatile unsigned int *) (0x400C0000) /*write a 2 to start convertion*/
#define ADC_ISR * (volatile unsigned int *) (0x400C0030) /*status reg -- bit 24 is data ready*/
#define ADC_ISR_DRDY 0x01000000
#define ADC_START 2
#define ADC_LCDR * (volatile unsigned int *) (0x400C0020) /*last converted low 12 bits*/
#define ADC_DATA 0x00000FFF

Then the sample loop becomes:

ADC_CR = ADC_START ;
  for (i=0; i<320; i++){
    // Wait for end of conversion
    while (!(ADC_ISR & ADC_ISR_DRDY));
    // Read the value
    analog_data[i] = ADC_LCDR & ADC_DATA ;
    // start next
    ADC_CR = ADC_START ;
  }
For TFT display connections, look at the Adafruit tutorial, and consult the source code above. I used the following with the line
//#define USE_ADAFRUIT_SHIELD_PINOUT in the file Adafruit_TFTLCD.h commented out. 
// The control pins for the LCD can be assigned to any digital or
// analog pins...but we'll use the analog pins as this allows us to
// double up the pins with the touch screen (see the TFT paint example).
#define LCD_CS A3 // Chip Select goes to Analog 3
#define LCD_CD A2 // Command/Data goes to Analog 2
#define LCD_WR A1 // LCD Write goes to Analog 1
#define LCD_RD A0 // LCD Read goes to Analog 0
#define LCD_RESET A4 // Can alternately just connect to Arduino's reset pin

// With shield defined (from pin_magic.h)
// LCD Data Bit :    7    6    5    4    3    2    1    0 (LCD data pin)
// Due pin #    :    7    6   13    4   11   10    9    8 (board pin #)

A minor rewrite displays the frequency of the periodic input wave every second.

the board