The Liturgy of the Hours (LH) - also called Divine Office (DO) or simply Office - is the daily, public, shared prayer of the whole Church. It is just as much "The Liturgy" as the daily Mass. We need to treat it with very nearly the same reverence and joy that we treat and dare to approach the Holy Eucharist, both in Holy Mass and in Adoration.
If we're praying it in private, maybe with our wife or our family, perhaps we don't need to be too punctilious about the externals - candles, vestments, standing and sitting and bowing, etc. But we should definitely have the right kind of spirit in praying it. We should never rush through it, just to get it done. We should never do it by rote, letting our minds wander already to the day's troubles etc. We should never forget we are praying for all those, both in the Church and outside, "as much for the living as for the dead," who may not have anyone else to pray for them.
This is one of the reasons I encourage you to learn to sing the DO, and why I mostly do it that way myself, even when alone. Singing forces me not to rush. Singing forces me to pay attention to my breathing, and to the words. Singing occupies parts of my mind that are drifting when I don't sing the DO. I've learned from experience what a lousy pray-er I can be, and I am grateful to the Church's wisdom for reminding me that the Psalms are always meant to be sung. It's easier to be recollected when singing the Office. (It's also more solemn, and more different from any other kind of public reading or recitation, etc, etc, just as it is with chant in the Mass.)
Singing also connects me and my offering of the DO more concretely to the whole Church. Maybe that's more subjective than I usually think it is, but the fact remains that the Church as a whole has always sung the Psalms. Generations of monks and deacons and pious husbands seem to be looking over my shoulder, or hovering just out of sight, when I'm singing the Office. The Church still recommends the DO be sung, as much as possible.
If we're leading LH in public, we should take the same care to prepare and make use of the externals as we do for Mass or Adoration. The proper vestments for leading LH are alb and stole (and cope); or cassock and surplice and stole (and cope). Four or six candles should be used, on or next to the altar. For more solemn events or days, there should be procession, incense, etc; for less solemn, there are simpler but still dignified ways of doing the same thing.
The LH combines very well with other liturgical rituals - sacraments outside of Mass, liturgy of the Word, and especially Exposition and Adoration (and Benediction).
There are rules about how to combine these different things, and in what possible combinations, and you can look those up - or, for you newer guys, ask me or one of the older guys.
Some ammo from the General Instruction on the Liturgy of the Hours (every Psalter and Breviary has this document in it):
15. In the LH the Church exercises the priestly office of its head, and offers to God "unceasingly" a sacrifice of praise...
16. When the Church offers praise to God in the LH, it unites itself with that hymn of praise which is sung in the heavenly places throughout all ages; it also receives a foretaste of the song of praise in Heaven...
18. Those who take part in the LH bring growth to God's people in a hidden but fruitful apostolate, for the work of the apostolate is directed to this end, "that all who are made sons of God through faith and baptism may come together in unity, praise God in the midst of the Church, share in the sacrifice and eat the supper of the Lord."
23. Those in Holy Orders [among others] have the responsibility of initiating and directing the prayer of the community... They must therefore see to it tha the faithful are invited -- and prepared by suitable instruction -- to celebrate the principle Hours in common, especially on Sundays and feast days.
255. The priest or deacon who presides at a celebration [of LH] may wear a stole over the alb or surplice; a priest may also wear a cope. On greater solemnities, there is nothing to prevent several priests from wearing copes, or several deacons from wearing dalmatics.
256. It is for the presiding priest or deacon, from the chair, to open the celebration with the introductory verse, to begin the Lord's Prayer, to say the concluding prayer, to greet the people, bless them and dismiss them.
258. In the absence of a priest or deacon, the one who presides at the Office is only one among equals; he does not enter the sanctuary, or greet or bless the people.
261. During the Gospel Canticle at MP and EP, the altar, then the priest [deacon] and the people may be incensed.