HW relics are probably the easiest relics to complete, and are easily completed without spending a single gil (except for teleporting, I suppose). A majority of the HW relic is just Poetic tomes, more and more and more and more Poetic tomes. You can do a lot of it solo, but you might need some help for a handful of things.
How long will it take?
That entirely depends on how fast you're generating Poetic tomes, since that's the bottleneck for many steps. You can actually prefarm a bunch of the stuff (maybe only if you've completed a HW relic before?). I like to buy all the poetics stuff in advance, then when I'm done I'll knock out a relic from start to finish in a day or two.
Step 1: Do some fates.
- Fast, easy, solo, free.
Step 2: Do these 10 old dungeons.
- Boring, but easy, soloable (unsync), and free.
- You can use a better weapon for the entirety of the dungeon, just make sure you have the crappy relic equipped when you exit.
Step 3: Get these various materials
- Easy, soloable, free.
- You can buy half of them with Grand Company seals. 80,000 GC seals in total.
- You can buy the other half with Poetics (and a handful of other methods, like ARR/HW beastmen quests). If you go Poetics-only, it's 6,000 Poetics.
Step 4: Get 5 Aether Oil
- Easy, soloable, free.
- You can buy these with 1,750 Poetics.
Step 5: Get a bunch of Crystal Sands and Umbrite
- Easy, soloable, free.
- The Umbrite will cost you roughly 4,500 ~ 5,500 Poetics.
- The Crystal Sands are most easily obtained via DoH/L Yellow Scrips by buying and trading in a bunch of Rowena's Token (Blue).
- If you don't do DoH/DoL, then some people run HW leves for the Amber-encased Vilekin trade-ins.
- You can technically get the sands via poetics as well (via the Unidentified items turn-in), but that's like... a shitload of poetics.
Step 6: Get 50 Singing Clusters
- Easy, soloable, free
- You can get these via a couple quests, or just buy all 50 for a total of 2,000 Poetics.
Step 7: Get 15 Pneumite, run 3 lv60 dungeons, get a bunch of light
- Easy, free, technically soloable but getting light solo is way slower than doing it in a group
- 60,000 GC seals for the Pneumite
Step 8: Get 1 Archaic Enchanted Ink, and run a bunch of old trials
- Mostly soloable, free
- 500 Poetics for the ink
- Most of the old trials are easily soloable. You might need a couple friends to tackle the last 4.
- You can use a better weapon for the trials, just make sure you have the crappy relic equipped as you exit.
Strictly adding up my bolded figures, that's...
- ~15,000 Poetics
- 140,000 GC seals