Criminal have been in a pretty weak position lately. Anarchs have been dominant, and it seems as though any attempt by the MWL to bring them back into line just causes Shaper to pick up the slack.
Temujin Contract, I think, is an attempt by the developers to bring Crim back. Because hoo boy is this card good.
This is, obviously, an economy card. And the value of an economy card is the number of credits it provides you above and beyond the things you would have been doing anyway. With that in mind, there are two ways to size up Temujin Contract. Similar to how Dirty Laundry works, you can make throwaway runs, or valuable runs.
If you make throwaway runs (on Archives, typically) then Temujin essentially becomes a turbocharged Liberated Account. Liberated Account gives you a net gain of 10 for an expenditure of 5, for an economic value of +5. Temujin Contract gives you a net gain of 16 for an expenditure of 6, for an economic value of +10, twice as good.
This is bonkers amounts of money, but there is a catch: the corp can cut into your profits by putting ice in your way. To avoid this, you'll probably want your Temujin turn to be "Install, run three times", which lets you get a lot of value before the corp has a chance to respond. In fact, you're making 8 credits over 4 clicks, which is 4 clicks on a Mopus or a Day Job, and you still have two 4-credit runs waiting for you if they don't ice up.
At this point, the corp will generally throw down something to stop you. For most pieces of ice, the cost to break it twice exceeds the cost to rez it: Enigma costs 3 to rez and 2 to break with a Passport, so by playing it they are reducing your Temujin payoff by 4 for a cost of 3 and a click. A more efficient taxing piece of ice like Eli 1.0 or Data Raven might even put them ahead.
And they know that you will, eventually, need to get into that server to clean off the rest of the Contract. Why? Because Temujin Contract is unique. So long as you have one on the board, you cannot play another one to replace it. So by blocking the completion of the first Contract, they are blocking you from playing the other two you probably have in your deck. And once they've iced Archives, the second Contract would need to find a new server to target. This gives them a pretty strong incentive to slow you down.
[EDIT: The previous paragraph is incorrect. The rule regarding unique cards is that if a second copy ever becomes active, the first copy is trashed. So you can forfeit any remaining credits by playing a new Temujin contract, although this is not generally optimal.]
This is all assuming, however, that you are running a server for no other purpose besides income. But Temujin Contract does not replace accesses. In fact, it places no requirements at all, besides "successful run on the chosen server". This means, then, than you can play the Contract targeting a server that is actually valuable, like R&D or HQ.
In this case, Temujin is essentially reducing the cost to get into that server by 4 for the next 5 runs you make on it. Since you aren't throwing away the run, this raises the economic value of the card from +10 to +15. But in doing so, you are again giving the corp an enormous incentive to ice the bejeezus out of the targeted server, not only to slow your economy down, but to stop you from getting those accesses.
But hey, this is Criminal. Forcing the corp to rez ice, forcing the corp to protect their central servers at the cost of their scoring remote, and making money by running, all of these are solid strategies that synergize super well with what Criminal is trying to do. If Gabe with a Desperado files a Temujin Contract on HQ, then Emergency Shutdowns the ice protecting it, the corp is going to have to scramble to keep it locked down. Meanwhile you can Account Siphon them, Legwork, them, and just when they think they've got it secured, you can play that Sneakdoor Beta and get in sideways. Tasty.
It's only 2 influence, though. Which is sad, because it means that it's probably not going to be played in Criminal decks as much as it really deserves. Instead, the Anarchs and Shapers will import it to fuel their late-game Keyhole/Medium/R&D Interface digs.
In summary, Temujin Contract puts a huge strain on the corp to either fortify the targeted server, or let you walk away with ludicrous dosh. A deck that wants to make money by running and that can capitalize on this pressure will love Temujin Contract.