what is the most economic and efficient way of going to Amsterdam city from Frankfurt airport?

None of the "low cost" carriers fly from A'dam to Frankfurt - so you are dependent on the normal carriers such as KLM, Lufthansa & co, and so unless there are special offers or booked well in advance, can be quite pricey. e.g. a single on 24th May which is direct to Frankfurt (no transfers) - the cheapest is €472 with Lufthansa http://www.ebookers.nl/afo/flights/step2.html?VERSION=NL&LANG=NL&from_wait=2140&SID=S C1oaQrfNGgAAEQVLI0&

One alternative is by rail. It takes 3 hours 40 mins and the proce varies depending on which day you travel but e.g. Sat 24th May a single ticket costs €39.00 http://217.194.102.76/cobs/app/microtips/display

Eurolines (coach) takes about 8 hours (again depending on when you travel) and for a single on 25th May it costs €42.00 https://webapp.eurolines.nl/eurolines/ssl_index.asp

It really does depend on when exactly you want to travel (e.g. by train it is generally more expensive during the week than weekend), and also if you want a return or single (single tickets being expensive by normal airplane carriers) so it is best if you check the links yourself for the dates you want to travel

P.S. I just realised I may have used the Dutch versions of the sites but all of them are in English as well, just click the flag or "English" text (usually top right) to change to English

Just take the train. There aren't budget flights that would save you any money. You can get the train right ther in Frankfurt airport.
For example, there's a train at 11:43 that arrives in Ams. at 15:25. There's one at 13:43 that arrives at 17:25. Those are the highspeed ICE trains, and they cost about 105 euros. But if you know your date and time, you can go online and look for special advance purchase fares that are as cheap as 39 euros. They are not refundable or changeable, though.
At Frankfurt airport you catch the trains at the "Fernbanhhof"
(Long distance train station). The ICE trains take 3 hours and 42 minutes to Amsterdam Centraal, but you can save some money and lose some time by taking the IC and RE trains that require some connections.

http://reiseauskunft.bahn.de/bin/query.exe/en?ld=212.125&seqnr=2&ident=nn.07495125.1210935406 &rt=1&OK#focus

Also try mitfahrgelegenheit.de. This is a ride-share board and you may find a ride for about 30€. Have fun....:o)

Train, I should think.