What are the different methods and their respective odds of getting/earning character specific items?

If you're referring to the 5* items that can allow a new soul break to be used when a specific character equips it (e.g. Hell's Gate when Sephiroth equips Masamune, Healing Wind when Aerith equips Guard Stick), those are only available through relic pulls. Note that 5* equipment is not limited to rare relic pulls as it is possible obtain any piece of equipment from the daily free common relic pull.

There are no offically published numbers on relic pulls (at least for Global), but here is the best information I was able to find on the probabilities:

Rare Relic:
3*: 75%
4*: 20%
5*: 5%

Common Relic (percentages rounded a little):
1*: 73.9%
2*: 20%
3*: 6%
4*: 0.08%
5*: 0.01%

(Source: https://www.reddit.com/r/FFRecordKeeper/comments/31lidm/comparing_the_rate_of_relic_pulls_of_japan_and/)