I'd say the best way to start is with a small skirmish game and then move up to bigger games as you get more figures painted/assembled. So you'd be following a trajectory that goes from small skirmish (5-10 figures a side) to large skirmish (dozens of figures a side) to mass battle (hundreds).
For small skirmishes, you can't go wrong with
Song of Blades and Heroes, even though it's ostensibly a fantasy game. It's cheap (not much over a fiver in PDF, I think, or under a tenner in print from Amazon), and it offers plenty of ways to individualise characters. So you'd have no trouble statting up berserks, huscarls and fyrdmen (and if you wanted to throw in a troll or a draugr, you could do that too). It works well for solo games because of the activation mechanism, which means that you can't always do what you want and your turn can end suddenly - making for lots of reactive choices when you switch sides.
Song of Blades and Heroes also has very simple mechanics but a wide range of outcomes from them - so you get knockdowns, pushbacks, kills and gruesome kills in melee. And it just plays very like a real fight - lots of backing away and losing nerve, rather than robotically staying stuck in. There are several historical variants, but there isn't a specifically Dark Age one because the fantasy rules cover everything you need for that period.
It's perfect for doing the kind of small-scale fight you get in the Icelandic sagas - homestead burning, cattle theft and ambushes and so on. And it's designed to have lots of non-combat objectives in scenarios. One of the supplements has a campaign system, but you could devise your own in seconds (winner gets to improve a characteristic for one figure, etc.). And the games are typically very quick - so that you can play through a campaign in an evening.
At the 'large skirmish' scale, there's Saga, as Steve said, which is great. It is expensive and somewhat complicated, though, given the battleboard element (you assign dice to a board to activate certain special rules), though easy to play once you've had a game or two. My one doubt about its solo viability would be the forward-planning aspect of the battleboards, which allow you to pull off surprises - but I'll defer to Steve on that.
I'd recommend going with Lion Rampant. It's a little late in period - Norman Conquest on - but the author has written some (minor) adaptations for the Dark Ages, and if you leave out crossbowmen and the heaviest cavalry types, it's all there. The troop types are broad (Elite Infantry, Heavy Infantry, Light Infantry, etc), and it assumes that most infantry are spear-and-shieldwall types. So it plays just fine as is.
The rules are very simple and fun. The most controversial aspect of the game is the loss of control (as with Song of Blades, your turn can end suddenly - or not even start), but that makes it good for solo play. Also, certain units have the Wild Charge rule, which takes them out of your control when they're close to the enemy. So there's the skeleton of a solo game there already.
The author, Dan Mersey, has also published the colonial-era variant The Men Who Would be Kings, which has a solo engine that you could port into Lion Rampant if required.
For bigger battles, I'd recommend DBA - an 'element-based' game. You'd need to mount your figures on temporary bases (or just group them together if they're on squares to begin with), but the rules give an absorbing game and have endless army lists for the period. The combat system is the ancestor of that used in Song of Blades, so if you've used that, the bigger game should come easily. And again, games are quick - maybe 45 minutes once you're used to the rules. Some people actually prefer the fantasy variant, Hordes of the Things, for historical battles; the rulebook is probably a bit easier to digest. The figure requirement for DBA/HotT is low, because three or four men on a single base are assumed to represent hundreds.