Equiptment: Only from looking at your current equiptment, I will tell you may be short on money. Some general modifications you could make are upgrading the Rune/Dragon armour to RuneScape gold Barrows armour, specifically Guthans (for set effect) or Torags (for exceptional defence bonuses). It is necessary to have boots/gloves/cape because every incentive counts. I'd buy Dragon Boots, or Rune, Rock Climbing, or Adamant. I would then purchase a Regen Brace, a Combat Brace or EVEN Leather Gloves, for the added bonus. To get a cape I would strongly suggest an Obsidian Cape, however a God Cape (out of Mage Training Arena), or any other cape will suffice. Personally, I would alter the Dragon Medium Helm to a Helm of Neitiznot, Berserker Helm, or even a Dragon Full Helm for extra, more useful bonuses. Also (this might be the most crucial thing)... Get an Abbysal Whip... Very costly, but really helpful.
I really don't have a lot of expertise at Barrows, so you will likely want someone else to enlighten you about the following: Order: The generally accepted sequence of murdering the Barrows brothers is: Dharok, Karil, Ahrim, Verac, Torag, Guthan. You will NEED to plead against Dharok, Karil, and likely Ahrim. You may also choose to pray against another three. Good Luck.
Right, for barrows, you need 80+ Attack/Strength/Defence, and 70+ Magic. 43 Prayer is a must but 70 is helpful for Piety. Black sallies require 70 in ranged, magic, and attack. Their ammunition is Harralander Tar and it is very effective, but can be rather costly depending on your magical level. Slayer dart requires 55 slayer, and it is principally used because when you loot the chest, you get a higher prospect of getting the runes for slayer dart ( 1 departure rune/4 mind runes per cast.)
Order of brothers: Dharok: you must use protect from melee. Use your magic attack. Karil: if you have leftover prayer, then utilize shield from missiles (Piety also helps a lot in this scenario). Switch to your melee weapon to OSRS Gold For Sale kill him. Ahrim: if you have leftover prayer, then utilize protect from magic. Equip D'Hide, and a crossbow or the shameful sally ranged style. Verac: if you've got leftover prayer, utilize it on keep from melee. Drink a Super Defence potion before looking for the sarcophagus. Take advantage of your mage design in great melee defence gear (Rune+). Guthan: use your mage style in melee armour. Torag: utilize your mage style in melee armour.