TL;DR Everything that goes wrong is your fault, learn from your mistakes rather than deflecting. I am working towards 4k MMR and there was no way in hell I would improve by blaming anyone other than myself. Even when your teammates go full retard, you are the only one in control of your choices.
First, you are not as strong a carry as you believe. You play almost exclusively DPS heroes, Nova and Hammer among the most frequent. I'll tell you right now that there are two kinds of players that get Nova to level twenty: 4k MMR players with a chip on their shoulder that dominate with Nova because they know they get away with it, and 2k MMR players who are ignorant of their own faults. You fall into the latter category and you will never improve unless you start doing some introspection.
Back to your carry problem. When you are not playing ranged carries like Nova and Hammer, you chose Illidan, Valla, Zeratul, Butcher, Tychus, and occasionally others. The fact that your posted image shows seven deaths on your part leads me to believe this is a 'typical' losing game. While there are always exceptions in the competitive scene, in HL your deaths should be few and far between. HL lacks coordination, so you know going in that you are entering a 1v5 with four other people that just happen to be allies.
Occasionally your allies may save you, but when that happens they are covering for a mistake that you made. Simply put, dying is your fault. Every time you die, you know for a fact that you were out of position. Analyze each death and figure out what you did wrong, correcting it for the next time you are in a similar situation. This is especially important in the early, pre-objective phases. If you end up solo laning your only job is to soak experience. Never ever, ever, ever, ever, ever is it acceptable to die while soaking a lane by yourself. If you die solo lane and lose that game, you also lose the privilege of complaining, even to yourself.
In addition, the 31k experience you gathered in your image indicates to me that you are not contributing enough to team fights. Yes, you have a large amount of hero damage but Hammer's range and AoE means that figure will always be padded. Split pushing is generally not an effective strategy, I often compare it to cheese tactics in Starcraft. If your makes a mistake or does not respond properly you get an easy win. However that strategy has less to do with personal skill and more gambling on your opponents lack of skill (this is the same issue Nova runs into). Everything is more effective as a team, whether it is teamfighing or pushing down a lane. If you want your team to push hard, explain to them how they should capitalize on winning a teamfight.
Second, tied closely to the first, is your lack of draft knowledge. In HL you are obligated to draft the best hero in the current situation. Being competent in every role is the bare minimum required to be competitive in HL. The fact that most players are not capable of this is why the minimum rank 1 MMR is so low (around about 2700). Not only are you not entitled to play only assassins, you are harming yourself by not learning the intricacies of tanks, bruisers, healers, and utility supports. Not only should you know your preferred heroes inside and out, you should be intimately familiar with the kits and capabilities of the heroes both allies and enemies are likely to draft.
A reminder, tier lists are not about power. Every draft pick is situationally dependent, higher tier heroes are those who's situation occurs more frequently than lower tier heroes. High tier heroes are more generally more versatile and consistent, which is why they are common. Some carries like Valla are 'one size fits all,' she is a safe pick in every situation. You favorites, however, are not. Hammer, for example, is highly situational. Personally I almost never draft her because it requires a degree of implicit teamwork. Voice coms are not strictly necessary but there is a specific strategy that everyone must adhere to. The last time I recall drafting Hammer was when a couple of my friends duo queued onto my own two-stack.