cerafino has already provided an excellent list of the individual upgrade pieces required for this achievement.
The full sets of the (purchasable) armor will cost approximately 150,000f. If you have spent money renovating the blacksmith's shop in Monteriggioni, you can save a good deal on those purchases. The first three pieces of the Missaglias armor become available around the time of the beginning of DNA sequence 9, and the final piece becomes available upon completing the sequence. However, the missions in this story line bind you to the city in which the missions are set, so you cannot take advantage of the savings in Monteriggioni if you want these pieces immediately.
The final hidden blade upgrade (pistol) also becomes available during this mission strand.
The large version of each of the pouches costs 6000 florins. If you purchase the large pouch before purchasing the medium (small) one, you can save the 3000 florins of the purchase price for the small pouch, and still get the achievement. The last of these upgrades also becomes available around the time of DNA sequence 9.
The fastest ways to accumulate wealth are to perform some renovations in Monteriggioni and wait, finding (and dispatching) Borgia messengers, or to perform assassination side missions. Borgia messengers yield 1500 florins, and assassinations are usually 2500.