The chanting of the psalms in the youtube video is on the special paschal tone. Normally Compline is chanted "in directum" in the Benedictine Office, which is the tone used for psalmody without antiphons. Traditionally, the Benedictines don't use antiphons at Compline, nor do they chant the repsonsory "in manus Tuas" and the Nunc Dimmittis. Those are features of the Roman rite though some monasteries now chant them (including the abbey I am affiliated with). Instead of the responsory the versicle "Custodi nos, Domine..."
In directum is another tone that can be used for the aspiring cantor, that is a wee bit more complex than recto-tono and requires the paying of attention to the Latin accentuation. It is simple, with an inflection of two notes each one half-tone lower than the other, returning to the recitation chord on the last accented syllable (if the accent is the second or third syllable from the end) or the second-to last syllable, if the last accent is the 4th syllable from the end).
Recto-tono is used in the Office. In the abbey I am affiliated with, it is used for the psalmody and readings of Matins, and also for the psalmody of the minor hours.
Compline is done in directum druing the week, and on a more ornate (in French) tone on Sundays. The advantages to in directum and recto-tono, is that they can also be easily adapted to the vernacular, which is not so easy for the traditional Gregorian modes except the Irregular Tone (one of the 4 archaic tones).
The paschal tone heard on the youtube video would be the next step up in complexity after in directum.
Then when you really get good you can hit the really tough tones like IIIa2 or IIIg :D
Some abbeys (and orders) do the entire office recto-tono.
For hymns, for the ferial office they are generally octo-syllabic, and can be done on a fairly easy tone (that normally used for the minor hours) to get used to chanting them.
My suggestion to chant an entire office would be to start with Compline in the Monastic style, in directum, with the simplest or Sunday setting for the hymn Te Lucis. Solesmes has a great recording for it (that also includes Sunday Vespers).