Your first problem is that many of the multi-day hikes in Iceland don't really become passable and/or accessible, until late June or even early July. Your next problem is that there is a large on-going volcanic eruption which no one can say when it will end, and it is likely to close off a large area which includes some important hikes in the vicinity of Askja.
Take into account that it should be a low budget trip.
Best avoid the two most expensive countries in Europe then because incidental expenses when not on trek, especially transport to the trailhead, will be expensive. But the best thing you can do to keep the costs of a trip to Iceland or Norway down is go on multi-day hikes, the longer the better, because you can't spend money on anything other than food or camping (or huts) while on hike. It doesn't really make very much difference which one you go on, though maybe you'd prefer the ones where you wild camp because then you avoid the overnight fees.
As to which are the 5 top ones, that is very subjective. I suggest you buy a copy of the Cicerone guide to hiking in Iceland and see what they say about it and form your own opinion from what they say. Also browse the websites of tour organisers who arrange multiday treks, since they try to put on the most exciting ones - though they do also tend to favour those of particularly tricky navigation, difficult access, hard river-crossings, etc, so that they are giving you value for paying for the guide. Also have a look at the 4 sketch maps of eastfjord trekking routes which can be found if you look hard for it on www.nat.is for some little known trekking routes which are definitely DIY.
Some points to bear in mind:
- all transport is expensive in Iceland, especially in the interior
- Hornstrandir hiking has the significant expense of the boat to get out there
- some routes start/finish in hard-to-get-to places, eg the Lonsoraefi trek has one ends at the Snaefell hut which is on a 4wd road, but a long way from anywhere with public transport or material passing traffic for a hitch; you might end up needing a v expensive mountain taxi to get you out if you haven't got the food to let you walk (much further) to a road with passing traffic. There's also the trek that ends at Svartakot (which may not be doable this year because of volcano); people do walk back out to Myvatn from there in a long day, but it's something like 40km.
A number of threads on alternative long hikes in Iceland are on the forum, have a browse, including links which clarify some of the above.
You should obviously spend as long as you can afford in Iceland as it has the most fascinating and diverse scenery in all Europe.
