I love a bit of soloing as well as small teams (and big teams for that matter, hell it's ALL good!).
IMO, soloing is where you learn how to play your character well. It's where you learn the tricks that will save your life (and maybe even prevent team wipes) throughout the game, so you're definitely not doing that bit wrong.
Firstly, don't bother with hero builders like Mids and Pines. You can always respect into fancy builds later on - your main priority is to fill the gaps in your attacks.
Secondly, when you're soloing, damage isn't everything. Stun attacks, knockbacks, immobilises, basically anything that stops the enemy from doing something, even for a few seconds, can be more valuable than an extra attack that doesn't take them down at a critical point in the fight.
Thirdly, manage your aggro. Open with a light damage attack to start with. If you're lucky you'll only pull one enemy out of a spawn, then you can draw him away and defeat him in isolation. If you're unlucky and pull the whole spawn, switch targets and use one of those KB/immobs/whatever powers to keep at least one of them out of the fight. It's always tricky to do this at low levels, but it's great practice for when you've got a few more powers on quick recharges.
Get used to switching targets. Prioritise your targets based on how much of a threat they are at any given moment. It's often best to finish off minions first so you can concentrate on lieutenants without taking hits from them, but it's always a good idea to keep the lieutenant knocked down, knocked back, held, immobed or otherwise engaged while you do. Again, pick powers that give you this utility.
If you get out of your depth, run. Try to keep an enemy targeted so you can see when they stop chasing. When they stop, rest, then go back to finish them off. Don't be too proud to leave the mission if that's what it takes. Retreating through lifts can work too, but sometimes you'll be chased through them so be careful.
Sell any enhancements you find that aren't going to last for long (which is most of them at early levels). The boost they give is negligible and you'll need to inf for DOs at level 12.
Don't get a travel power at level 4 unless you feel you absolutely have to. That power slot is probably better used on another attack to get your rotation working. The Ninja Run you can get from the P2W vendor is pretty serviceable as a travel power at low levels, but it looks cheesy as hell. Depending on your character concept, that might be a good enough reason not to get it ;)
Even though you hate melee, it never, ever hurts to have a melee attack on hand for emergencies. You don't necessarily have to slot it, but it can be really useful for finishing off an enemy before retreating.
Also, pick your contacts and missions carefully - you will be vulnerable to some enemy types and strong against others. Play to your strengths. Circle of Thorns are my particular bugbear, but your mileage will vary.
There are loads of little tricks you'll pick up as you play, and they'll vary from AT to AT and powerset to powerset. You'll also develop a feel for you character's powers, how they synergise and how the timings fit together.
You will die, but you'll die less often as you get that instinctive feel working for you.
When you do die, restock your inspirations. Heals are always good as are defence insps. In the high teens, don't forget to keep a few accuracy insps on hand - there's a low-level accuracy bonus that drops off as you approach level 20, so they'll be useful if you haven't managed to slot for accuracy.
Slotting varies from power to power (and you'll find your own preferred slotting as you go), but early on it's good to slot for recharge so you've got attacks available more often. As you get to higher levels endurance can become an issue, but you can probably fix that as it starts to happen. Mid- to late-teens, don't forget to slot that accuracy.
These are some of the tricks I've learnt from hundreds of levels of soloing and small teaming with all kinds of ATs, so hopefully they'll help.
One last thing - be patient. Pick your targets, don't be afraid to run when you need to, and don't be perturbed by dying early on. Once you've got the feel for you hero, you'll die a lot less. You'll figure it out.
