DSolve doesn't seem to want to do this, but NDSolve can do this.
In[1]:= m = 1; k = 1; l = 1;
sol = NDSolve[{\[CapitalTheta]'[t] == l/(m r[t]^2),
m r''[t]^2 - l^2/(m r[t]^3) + k/(r[t]^2) == 0,
\[CapitalTheta][0] == 0, r[0] == 1, r'[0] == 0},
{\[CapitalTheta][t], r[t]}, {t, 0, 10}]
Out[2]={{\[CapitalTheta][t]->InterpolatingFunction[{{0.,10.}},<>][t],r[t]->InterpolatingFunction[{{0.,10.}},<>][t]},
{\[CapitalTheta][t] -> InterpolatingFunction[{{0.,10.}},<>][t],r[t] -> InterpolatingFunction[{{0.,10.}},<>][t]}}
In[3]:= Plot[{\[CapitalTheta][t], r[t]} /. sol[[1]], {t, 0, 10}]
<<<plot snipped>>>
In[4]:= ParametricPlot[{\[CapitalTheta][t], r[t]} /. sol[[1]], {t, 0, 10}]
<<<plot snipped>>>
But neither sol[[1]] nor sol[[2]] seem to show the circle that you describe. Is there perhaps some error in this?
Edit: I think this is what I should have understood to write.
In[5]:= ParametricPlot[{r[t]Sin[\[CapitalTheta][t]],r[t]Cos[\[CapitalTheta][t]]}/.sol[[1]],{t,0,10}]
Note: Hopefully I haven't introduced any errors by having to manually type in the contents of that second code box, because apparently the forum software makes it impossible to paste into a second code box. Or at least that is what appeared to have happened repeatedly for me.
It also seems surprising that DSolve cannot find a solution r=1, theta=t to the system of equations, with or without taking the liberty of multiplying both sides of both equations by an appropriate power of r to clear the denominators.
