How does one evaluate a hand responding to a "strong" two clubs?
The system I have played for many years is as following:

1. Opener rebids NT to show a balanced hand with a well defined point range:

Responder evaluates his hand as balanced unless making a transfer bid into his own suit. These auctions are very similar to 2NT openings except with greater strength having been shown by Opener.

1a. Opener rebids 2NT with a semi-balanced hand of appropriate strength (usually a range of 22-23 HCP) and a 5-card major (not longer) with an agreement enabling Puppet Stayman. Advantage is that Responder's second negative doesn't place Notrump contracts in his hand with the strong hand exposed and led through.


Responder will almost always rebid 3C (Puppet Stayman) to allow Opener to reveal a hidden 5-card major (or 6-car minor).

2. Opener makes a simple rebid in his Major suit:
Responder bids 2NT as a second negative with either less than 3-card
support or 3 cards only and a bust hand.
Responder raises to 4 of Opener's major with 3 or 4 card support and a few scattered points.
Responder raises to 3 of Opener's major with 3+ card support and slam interest, but no outside control.
Responder bids a new suit to show a control (either A or K) in the suit bid, 3+ card support for Opener's major, and no reason to deny slam interest.
Responder jumps in a new suit to show 4+ card support and shortness in the bid suit with no reason to deny slam interest. Generally this call is preferred to the simple cue-bid, as controls can be asked for later.

3. Opener makes a simple rebid of his minor suit:
Responder rebids as above, but with the change that all bids above 3NT emphatically deny interest in 3NT (obviously). 
Responder rebids 3NT with a balanced 7 or 8 count well scattered and lack of interest in slam. The inference is that responder believes 9 or 10 tricks are available through sheer power, but that the fit is insufficient to make slam realistic. Opener continues only if a) his hand is completely unsuitable for notrump; or b) his hand requires only 2-card support to play well in his minor.




Miscellaneous comments on the situation:

It is vital to remember that will not be Captain for the auction will not be the 2C opener but rather responder. The key descriptive features of a 2C opener are that it is a strong hand, and can be described adequately in 2 calls so that partner can be captain. More complex hands should not be opened 2C, and both members of the partnership needs to be aware of that.

As such, all attempts at step and control responses to the 2C opener have been abandoned by serious players (with the possible exception of pure relay systems, which you cannot play in most tournament events). 

For instance, the hand xxxxx-xxxx-x-xxx can be golden if opener's suit is spades with three worthless diamonds, and garbage if opener's suit is diamonds with three worthless  hearts. No bidding system can hope to describe the range of partner's hands adequately to opener, especially  when that conversation has to start at the three level, which is why responder is the captain as soon as the 2C opener is made, until and unless the opener can glean enough information to bid Blackwood.

Update 

One of the ways in which opener becomes captain if he has support for a quality 5-card suit (or longer) shown by responder as a positive response. When this happens opener sometimes is in a better position to judge the combined playing capability of the hand by bidding Key-Card Blackwood in responder's suit.

Update #2:

Only the captain can ask for key-cards, because the premise is that the key-card responses will allow the asker to place the contract, both denomination and level, and only the captain can make that call (or make the requisite invitation). 

Opener can take over as captain only when responder leads opener to believe that his (responder's) hand is fully described - there are circumstances where it may be appropriate for responder to have a stronger hand and/or a different fit in mind than he has lead opener to believe, but that is an advanced bidding technique that intermediate players have difficulty handling.

Also, it is critical that all responder's calls while opener believes himself captain be in sente rather than in gote, so that responder will have the guaranteed opportunity to make the final correction.  This can require responder to have considerable foresight in planning the auction.