Short answer: Buy the What Lies Ahead data pack. That with the core set lets you build two legal runner decks and two legal corp decks.
I assume from your question that you've already read the rules on deck building and aren't just trying to build the starter decks. If you're thinking about buying data packs, you pretty much have to be building custom decks for things to even make sense.
If you're just going to play you're own decks against each other, you can do whatever you want in terms of deck building. When limited in cards, ignoring influence limits is the best way to be able to follow agenda density requirements and deck minimums. As long as your decks are fun to play against each other, that's really what matters.
If you want to follow all the deck-building rules, here's the run down:
The base game comes with enough cards for you to be able to build two legal runner decks by having each run cards from the third faction. Remember that you can spend up to 15 influence on out of faction cards (at least for the core set identities). For the runners, all they need to do is get up to 45 cards (at least for the core set identities) while still staying under influence limits.
Let's assume we're making Shaper and Anarch decks. The core set comes with 31 shaper cards and 32 anarch cards, with 15 neutral cards. If we split the neutral cards between the two, that means we have two decks of 39 and still haven't used any influence. That means each deck can run 6 criminal cards and has 15 influence to spend doing so. Between Forged Activation Orders, Special Order, Easy Mark, Femme Fetale, and Bank Job, (all either 1 or 2 influence per copy) you're good to go.
The primary limitation in building decks is the corp's agenda requirement. The core set does not come with enough agendas to legally build multiple corp decks. This is problematic because agendas cannot be run in decks of other factions (they have no influence value, which is different than having an influence of zero). Of secondary concern for the corp is ICE. The rule book suggests 17 - 20 pieces of ice for a 45 - 49 card deck. This is an area where the corps might be hurting.
The deck building rules require 20 - 21 agenda points for the 45 - 49 deck size required by all the core set identities. Given the core set only ships with 5 or 6 points of agendas in each faction and 15 points of neutral agendas, we need more to work with here.
The data pack What Lies Ahead is the best of the data packs for enabling more decks because it contains an agenda of each faction. If you only buy this data pack, one of your corp decks must be NBN, as you need the 3-point Restructured Datapool. This lets you run the neutral Private Security Forces in your NBN deck and the Priority Requisitions in your other corp deck of choice (along with the core set agendas and the corresponding What Lies Ahead agendas) and have two corp decks that meet agenda density requirements.
Now on to the ICE issue. NBN (which must be one of your corps because of agendas) starts with 9 pieces of ICE from the core set, and gain another 3 (TMI) from What Lies Ahead. If we split the core set's 6 pieces of neutral ICE (Wall of Static and Enigma) between our two corp decks, that means the NBN deck is already up to 15 pieces of ICE. This means you only need 2 - 5 pieces of out-of-faction ICE which is fine on a 15 influence budget (look at Ice Wall, Shadow, Rototurret, Chum, and Wall of Thorns for 1-influence ICE). You also get the neutral ICE Draco in What Lies Ahead, which probably works well in the NBN deck (goes with the "trace" theme).
All of the other corporations have at least 9 pieces of ICE in the core set, get another three from What Lies Ahead (though if you're running Haas-Bioroid, you'll probably need a Priority Requisition or an Accelerated Beta Test to be able to rez Janus 1.0, so maybe don't run all three copies of him), which leaves you in a similar situation to the NBN deck. Pretty easy to get to the 17 - 20 range.
The fact that What Lies Ahead includes a piece of ICE and an Agenda for each of the four corporations is what makes it the ideal data pack to buy if you're just going to buy one. Also, Morning Star is amazing.
I'm curious as to how you're playing 2x2 Netrunner, as the standard rulebook does not support this at all. The only rules I've run into on the topic were designed for the old Netrunner game (that's Netrunner the CCG, not Android: Netrunner the LCG), but could probably be adapted: http://www.darkpact.de/netrunner/variants/theBigSellOut.html. That said, the simple solution is just pair off and play two games at the same time. I'd be interested to know which you do.
A Purchasing Alternative
All you need from the core set to build multiple starter decks is a set of each of the neutral cards from the core set. There are ways of buying these sets, such as buying a special of all the "triplicate cards from the core set" from http://borntoplaygames.com/product-category/customizable/netrunner/. This would let you use the rulebook's recommended starter decks and have any corporation pair you want.