You need a phone that is 1) unlocked, 2) has the correct radios and 3) supports the correct bands. For radios, both Verizon and T-Mobile use LTE for high-speed data, but Verizon uses CDMA for voice and low-speed data while T-Mobile uses GSM. They are both transitioning to LTE for voice as well but it will be some time before that's the standard nationwide. Most new Verizon phones support GSM as well for the sake of being compatible overseas.
Band support gets a little trickier. Verizon uses Band 13 at 700MHz for the bulk of its LTE network, but is using Band 4 for its "XLTE" service. They are also adding Band 2 for additional bandwidth. T-Mobile uses Band 4 primarily and supplements with Band 2. They are also building out Band 12 but that's in the early stages right now. What that amounts to is that newer Verizon phones that are compatible with XLTE will work with LTE on T-Mobile, but older phones (even just a couple years old) will not.
The iPhone 5C is compatible with all of T-Mobile's bands (except for Band 12, which I wouldn't worry about right now) per GSMArena below.
Apple iPhone 5c - Full phone specifications