There are two types of 6th chords. The major 6th and minor 6th. Both of these chords are based on triads.

The major 6th chord (eg. E6) is based on the major triad. Once you have built the major triad from the major scale you add on the major 6th. If we are using E as an example this note would be C#. The E6 chord would therefore be E G# B C#. Here is the chord in tab...

E6
e|---0-E---|
B|---2-C#--|
G|---1-G#--|
D|---2-E---|
A|---2-B---|
E|---0-E---|

The minor 6th chord (eg. Em6) is based on the minor triad. Once you have built the minor triad you add on the major 6th. If we are using E as an example this note would be C#. The Em6 chord would therefore be E G B C#. Here is the chord in tab...

Em6
e|---0-E---|
B|---2-C#--|
G|---0-G---|
D|---2-E---|
A|---2-B---|
E|---0-E---|

So remember, for both the major and minor 6th chords, you add on the major 6th to the major or minor triads.